Measurement-driven navigation in many-body Hilbert space: Active-decision steering

Yaroslav HerasymenkoInstituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

Igor GornyiInstitute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

Institut fur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany andIoffe Institute, 194021 St. Petersburg, Russia

Yuval GefenDepartment of Condensed Matter Physics, Weizmann Institute of Science, 7610001 Rehovot, Israel

The challenge of preparing a system in a designated state spans diverse facets of quantum me-chanics. To complete this task of steering quantum states, one can employ quantum control througha sequence of generalized measurements which direct the system towards the target state. In anactive version of this protocol, the obtained measurement readouts are used to adjust the proto-col on-the-go, with a possibility for accelerated performance or fidelity increase. In this work, weconsider such active measurement-driven steering as applied to the challenging case of many-bodyquantum systems. The target states of highest interest would be those with multipartite entangle-ment. Such state preparation in a measurement-based protocol is limited by the natural constraintsfor system-detector couplings. We develop a framework for finding such physically feasible cou-plings, based on parent Hamiltonian construction. For helpful decision-making strategies, we offerHilbert-space-orientation techniques, comparable to those used in navigation. The first one is totie the active-decision protocol to the fastest accumulation of the cost function, such as the targetstate fidelity. We show the potential of 9.5-fold speedup, employing this approach for generating theground state of the Affleck-Lieb-Kennedy-Tasaki spin chain. The second wayfinding technique isto map out the available measurement actions onto a Quantum State Machine. A decision-makingprotocol can be based on such a representation, using semiclassical heuristics. This approach hasadvantages and limitations complementary to the cost function-based method. We give an exampleof a W-state preparation which is accelerated with this method by a factor of 12.5.

I. INTRODUCTION

Quantum state preparation is a prominent routine inquantum information processing toolbox [1–15]. Suchprocedure often implies steering a quantum system froma “simple” towards a more complex, pre-designated re-sourceful state, e.g. a many-body entangled state. Asteering protocol is characterized by an as short as pos-sible runtime and high resulting overlap with the targetstate. Constructing such protocols can be done in mul-tiple distinct ways. One is to design the Hamiltonian ofthe system, such that its unitary evolution leads to a des-ignated state. This paradigm is represented by methodslike digital computation or analog simulation [3, 5–8].Such protocols require exact knowledge of the startingstate, as well as the precise timing of the unitary evolu-tion, to be accurate. Another strategy is to make use ofthe environment, adding a dissipative element to the evo-lution. Combined with the Hamiltonian evolution, thisresults in methods such as drive-and-dissipation [11, 12].Finally, one can design a sequence of generalized mea-surements, which brings the system towards the targetstate via measurement back-action alone [16–18]. Therelevant part of the evolution is then completely governedby the system-detector coupling (see also Ref. [19]).Unlike protocols involving pre-defined unitary evolution,such measurement-driven state preparation may not re-

quire knowledge of the starting state and fine-tuning ofthe system Hamiltonian [18].

The above types of state-preparation strategies can bereferred to as passive, meaning that these protocols arepre-determined and pursued regardless of how the systemevolves. Given this perspective, it appears beneficial togo beyond the forms of control described above, and in-troduce the concept of active decision making. This typeof steering exploits information extracted during the sys-tem’s evolution to decide on the operations that follow.This is also referred to as closed-loop quantum controland is typically used to improve the Hamiltonian-basedstate preparation [21–26]. In that case, extracting thenecessary information requires the introduction of mea-surements into the protocol, which may result in an unde-sired back-action. Nevertheless, in certain cases, closed-loop control of Hamiltonian evolution does yield an im-provement to the speed and the fidelity of the protocol.

Another possibility, which is a subject of increasinginterest, is to employ active decision in measurement-driven protocols (implying no Hamiltonian drive) [27–29]. In such protocols, the necessary information aboutthe system is naturally available from the employed mea-surements. The active decision is then being madeabout possible changes in the subsequent generalizedmeasurements, such that the target state is prepared asrapidly and accurately as possible [17, 27–29]. Some gen-

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eral theorems have been stated concerning such activemeasurement-driven state preparation [29], along withsome specific protocols designed to reach single-qubit tar-get states [27, 28]. However, it remains unclear howan active measurement-driven protocol can be effectivelyharnessed to engineer resourceful many-body states. Inthis case, the large size of the Hilbert space makes itchallenging to actively steer the system evolution in thedesired direction.

In this work, we establish a general framework formeasurement-driven active navigation in Hilbert spaceand construct active-decision protocols for measurement-only steering of many-body states. In particular, we fo-cus on states manifesting genuine multipartite entangle-ment. When attempting to address the problem, one isnaturally constrained by a few factors. One is that onlyreasonably local system-detector couplings are to be usedin the protocol. Moreover, we require that the number ofdistinct system-detectors couplings available for steeringdoes not scale up faster than the system’s size (this num-ber should not be super extensive). This natural require-ment restricts the capabilities of the protocol and resultsin the need of correlating different system-detector cou-plings. Another prerequisite is that applying one typeof coupling generally leads to an update in the expectedbenefits from other couplings. This phenomenon, whichwe refer to as “coupling frustration”, calls for nontrivialcoordination between different coupling applications. Fi-nally, there is a problem of orienteering: it is relativelyeasy to “get lost” in the many-body Hilbert space whenexploring it with the set of tools limited by locality andextensivity (cf. Ref. [22]).

We note that upon the availability of indefinite com-putational power, one can always find an optimum se-quence of measurements through dynamic programmingtechniques (cf. Ref. [29]). Roughly speaking, this can bedone by considering all possible future quantum trajecto-ries of the system. However, in a large Hilbert space, it ispractically impossible to realize the theoretically optimalfeedback policy. This is because the extensive considera-tion of outcome scenarios is too complex for a many-bodyHilbert space of an already not very large system (it in-creases at least exponentially with the system’s size andthe duration of the protocol). Instead, we aim at design-ing heuristic strategies for active decision-making, whichwould allow for a significant – but not necessarily optimal– speedup of the protocol.

To meet these challenges, we introduce Hilbert-spacenavigation techniques. The first technique, which weterm greedy orienteering policy, is based on the notionof a cost function. A simple example of such a cost func-tion could be the target state infidelity. Minimizing it in agreedy protocol may already yield a reasonable advantagecompared to the passive policy. To test this approach,we study numerically the preparation of a ground state ofAffleck-Lieb-Kennedy-Tasaki (AKLT) spin-1 model [30].The numerical study shows the speedup factor that in-creases with system size, reaching factor 9.5 for N = 6.

Looking ahead, we discuss the fundamental challenge oflandscape flatness that may arise for some target stateswhen using simple infidelity as the cost function. Al-though this issue did not arise in the example we con-sidered, we propose a possible modification to the costfunction which should remedy this problem if it occurs.

The second technique is to map the Hilbert space ontoa colored multigraph, referred to as the Quantum StateMachine. The vertices of such a graph correspond to thebasis states, and the edges represent the actions of gener-alized measurements. Upon an appropriate choice of ba-sis states, such Quantum State Machine representationsallow for improved navigation in Hilbert space. Thiscan be done by heuristically representing it as quantumwayfinding on the graph. To substantiate this heuristic,we introduce the notion of semiclassical coarse-grainingof a Quantum State Machine graph. Optimizing the ex-ploration of these graphs by choosing the most appro-priate system-detector couplings results in advantageousactive-decision protocols. To exemplify this navigationparadigm we consider the preparation of the 3-qubit W-state, with a numerical study demonstrating a 12.5-foldimprovement in protocol runtime.

Throughout the paper, we assume that we know theinitial state of the system. This can be a “cheap” (say,product) and robust quantum state that does not requiremany resources for its preparation. However, one can di-rectly generalize the above approaches to the case wherethe initial state is unknown and is therefore representedby a density matrix. In the more intricate case of a Quan-tum State Machine-based policy, one would then need totake a weighted combination of graph navigation proto-cols with different initial states.

The remainder of the paper is organized as follows. InSec. II, we introduce the basics of measurement-inducedsteering. Specifically, in Sec. II A, we define the steeringprotocols and their elements, as well as the quantitativemeasure of the protocol’s success. Then, in Sec. II B, weillustrate these definitions as applied to passive steeringof a single qubit. The general selection criteria, includ-ing locality and extensivity, for the system-detector cou-plings, which are to be used for the active steering, areaddressed in Sec. II C. In Sec. III, we introduce the notionof frustration of steering and discuss the possibilities ofprotocols’ speed-up for mutually commuting (Sec. III A)and non-commuting (Sec. III B) couplings. In the lat-ter case, we develop a “parent-Hamiltonian” approach.A “quantum-compass” approach to active-decision steer-ing, based on the greedy cost-function accumulation pol-icy, is developed in Sec. IV, where we also employ thisscheme to the preparation of the AKLT state. In Sec. V,we develop an alternative active-steering framework – a“Quantum State Machine.” In Sec. V A, we introducethe generalities of this approach based on the underly-ing representation of the steering protocol in terms ofa quantum graph. Next, we discuss the quantum partsof this graph (Sec. V B), as well as the coarse-grainingprocedure, with the resulting coarse-grained graph being

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semiclassical (Sec. V C). This type of Hilbert-space orien-teering is illustrated in Sec. V D, where an active-decisionsteering protocol for preparation of a three-qubit W-stateis presented. Our findings are summarized and discussedin Sec. VI.

II. MEASUREMENT-DRIVEN STATEPREPARATION

A. Generalities

Measurement-driven state steering is a specific class ofstate-preparation protocols. Its basic building blocks arecoupling the quantum system (s) to quantum detectors(ancillary systems) utilizing engineered interactions, fol-lowed by strong measurement on the detectors (d). Thegoal of designing a measurement-based steering protocolis to generate a process that prepares the desired sys-tem state by utilizing a sequence of measurement back-actions.

Here, we will additionally assume that the internal evo-lution of the system and the detector are trivial (theirHamiltonians are kept null: Hs = 0, Hd = 0), as inRefs. [27–29], so that the unitary dynamics in the prob-lem is governed solely by the coupling between the systemand detectors determined by Hamiltonian Hs,d. For con-creteness of analysis, we also constrain the detector to bea qubit initialized in a trivial state |0〉, and the systemto be represented by N spins. Although a general spinS can be considered, we focus on the cases S = 1/2 andS = 1. We assume certain knowledge about the initialstate of the system, which is described by the initial den-

sity matrix ρ(in)s . For the sake of simplicity, we further

address the target state which is a pure state |ψ0〉.Although the ensuing protocol can be further general-

ized (see Section VI), we now formally fix its structureas given below:

Definition 1 A measurement-driven state steering is aprotocol that is performed to prepare a state |ψ0〉, starting

from the state ρ(in)s . It runs by repeating iterative cycles

of the following form (see Fig. 1):

1. Prepare the detector qubit in the state |0〉.

2. Based on the available information, select thesystem-detector coupling Hamiltonian Hs,d to beused in the next step.

3. Perform a system-detector evolution governed bya Hamiltonian Hs,d for a short time interval δt:Us,d = e−iHs,dδt.

4. Once the system-detector evolution is over, projec-tively measure the detector qubit in the Z-basis.Store the readout r for further processing.

5. Decide whether the protocol is to be continued orterminated. In the former case, return to step 1.

Measuredetector

System-detectorinteraction

Entangled stateafter interaction

Running state

Process

Decide on

StoreInstruct

Lab

Controlunit

FIG. 1. Basic design of the measurement-driven state prepa-

ration. The procedure starts with a given initial state ρ(in)s

and proceeds with a protocol, as described in Def. 1, until agood accuracy of the target state |ψ0〉 is achieved. The controlunit decides on the system-detector interaction unitary Us,d

based on the stored record of detector readouts. We focus onconstructing an optimized policy for decision-making, suchthat the target state is simulated as efficiently as possible.

Now, in the vast space of protocols that have such struc-ture, we would like to emphasize the distinction betweentwo classes of protocols: passive and active.

In a passive protocol, the stored readouts {r(t)} fromstep 4 may influence the decision for protocol termina-tion or continuation at step 5, but not the choice for theinteraction Hamiltonian Hs,d made at step 2 in the nextprotocol cycles. HamiltoniansHs,d can still be chosen dif-ferently for different iterations: e.g. for a large system,the detector qubit can be coupled to different subsystemsthereof. However, Hs,d used at each cycle in the passiveprotocol has to be pre-determined from the outset. Ifa passive protocol also has a pre-determined duration(and thus doesn’t use readouts {r(t)} at the terminationstep), we land in a subclass of passive protocols where thereadouts don’t have any influence on the protocol. Wewould refer to such protocols as “blind steering”. Forblind steering, the readouts of the detector at any givenstep can be averaged, i.e., following the measurement,the detector’s density matrix is traced out. In this work,however, we will focus on the non-blind version of pas-sive steering, where readouts are indeed employed for aninformed protocol termination.

In contrast to passive protocols, in an active protocolone uses the readouts {r(t)} to make an informed decisionfor the interaction Hamiltonians Hs,d as well as for ter-mination/continuation of the protocol. Active decision-making has to follow a certain policy, which becomes thecrucial part of the protocol. For a good active policy,its adoption should result in a significant speedup of theprotocol compared to its passive counterpart. Alterna-

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tively, one can also fix the protocol runtime and aim toimprove the precision of the state preparation. We fo-cus on the former target: minimizing protocol runtimefor a fixed target precision. The major challenge in thiswork is to construct such advantageous active decision-making policies. By comparing active steering with the(non-blind) passive steering as defined above, we inves-tigate the advantage offered specifically by the directedevolution, i.e. active decision-making for Hs,d.

Before we move on to the issue of active policy con-structions, let us discuss the criteria for termination of arunning protocol. In general, one cannot guarantee “per-fect steering,” i.e., obtaining the desired target state withthe fidelity of 1 in a finite number of protocol cycles. In-stead, one may consider preparing the target state withinfidelity R:

R(ρ(fin)s , |ψ0〉

)≡ 1− 〈ψ0|ρ(fin)

s |ψ0〉, (1)

where the state ρ(fin)s is the final state of the system once

the protocol is terminated. It is worth emphasizing thatthe system evolution during the protocol is probabilisticand depends on the stochastic readouts {r(t)}. It fol-lows that different runs of the same protocol may yield

different ρ(fin)s and, thus, the infidelity R. Therefore, to

characterize the protocol as a whole, we introduce thefollowing accuracy measure:

Definition 2 We refer to a measurement-driven state-preparation protocol as ε-precise, if the infidelity betweenthe final state and the target state is bounded by ε for anyrun of the protocol:

R(ρ(fin)s , |ψ0〉

)< ε. (2)

Given the knowledge of the readout sequence, we maysimulate the quantum system state (the quantum tra-jectory) on a computer in parallel to the measurementrun. Thus one can infer the running system state exactly(referred to as filtering in the literature [21]), and testinequality (2). This sets a trivial criterion for protocoltermination, which we will apply by default to all passiveand active protocols considered in this work. Namely, aprotocol can be terminated right after the cycle whenthe target state infidelity becomes smaller than ε, thusmaking it an ε-precise protocol. Apart from controllingthe precision, we are interested in the number of cyclesNc, after which the protocol has been terminated. AsNc may also differ greatly, depending on a specific run,we will characterize the protocol by 〈Nc〉run, where theaveraging is performed over many runs. Note that hereaveraging is taken over stochastic readout sequences. Inreality, steering errors as well as external noise may befurther contributing factors to the stochasticity. For thegiven target state and target precision ε (cf. Definition2), our goal is to find an ε-precise protocol such that〈Nc〉run is as small as possible. We will be consideringthis minimization as the key goal of our constructions.

B. Passive steering: Single qubit

As a simple example of a measurement-driven proto-col, we consider single-qubit steering (for a more gen-eral consideration, the reader is referred to Sec. II C).For simplicity, we will assume the target state to be|0〉, and the starting state to be a perfectly mixed state:ρstart = diag(1/2, 1/2). A single coupling suffices to guar-antee the preparation of the target state (in fact, from anarbitrary starting state) with an arbitrary precision [18]:

Hs,d = γσ−s σ+d + H.c. (3)

Here, σs and σd are Pauli matrices acting in the systemand detector spaces, respectively. By construction, a pro-tocol that operates with only a single coupling Hamilto-nian Hs,d, i.e., without a readout-based option of choos-ing different couplings, is considered passive. Neverthe-less, even for passive protocols, one can introduce a pol-icy based on the measurement outcomes, which wouldaccelerate quantum-state steering.

Let us first address a protocol that runs for N(pass)c

cycles using the coupling (3), regardless of the measure-ment outcomes. Under the definition given in Sec. II, thiswould be an example of blind steering. In this case, theprobability of obtaining a readout r = 0 decreases ex-ponentially with the total number of cycles Nc. Tracingout detector outcomes (since we are blind to measure-ment outcomes), this results in a density matrix:

ρ(Nc) =

(1− e−Ncγ2δt2/2 0

0 e−Ncγ2δt2/2

). (4)

Given the threshold infidelity ε, we need to run the pro-

tocol for N(pass)c (ε) cycles:

N (pass)c (ε) =

1

γ2δt2log

(1

2ε

)(5)

This characterizes the efficiency of the completely blindpassive protocol [18] for the single-qubit setup.

Next, we consider passive protocols where the sequenceof readouts is recorded. One then needs to interpret themeasurement outcomes, which for this setup is straight-forward. We note that when the readout is r = 1 (clickevent), the target state is instantly prepared [cf. Eq. (9)].Therefore, one can terminate the protocol directly af-ter the detector clicks for the first time: in this case,all further steps are simply redundant and do not re-sult in any evolution of the system. This will constitutea termination-policy improvement of the passive blindprotocol for this single-qubit case. If r = 0, i.e. noclick is measured (such a null-measurement [31] eventstill gives the system a nudge towards the target state bymeasurement back-action), the protocol simply continues

until a certain maximal number of cycles, N(max)c . The

target infidelity ε would be directly related to N(max)c

in a way equivalent to the blind protocol runtime (5).

5

The average runtime of the non-blind passive protocol,

N(pass)c ≡ 〈Nc〉run, is then given by

N (pass)c =

1

2γ2δt2

(1− e−γ

2δt2Nmaxc

)+Nmaxc

2. (6)

This runtime is strictly smaller than the runtime forthe passive blind protocol, Eq. (5), and yields a twofoldspeedup in the ε → 0 limit. It is worth emphasizing,however, that the termination policy can realistically beapplied only to the case of few-body quantum states. Forsuch a policy to be useful, a single detector click shouldsignify that the system is in the target state. This canonly be realized when the detector is coupled to all el-ements of the system. For a many-body system (manyqubits), a natural assumption of locality rules out such acoupling: a click of the detector coupled to a subsystemof the system does not guarantee that the whole system issteered to the desired state. Nevertheless, the above sim-ple example shows that detector readouts can be used foraccelerating the state preparation. In what follows, wewill focus on active feedback strategies. There, insteadof protocol termination, the local-measurement outcomesare employed for choosing the most efficient sequence offurther measurement cycles.

C. Selection criteria for system-detector couplings

1. Families of system-detector couplings

Both for the active and passive protocols, a key featureis the choice of coupling Hamiltonians Hs,d. Given thetarget state |ψ0〉, it is natural to constrain this choice toa certain family {Hs,d(p)}, for a set of (discrete or con-tinuous) parameters p. Deciding on the choice of Hs,d

in each protocol cycle translates into selecting the valueof p. Before discussing the policies for doing so, we ad-dress a different question: How to effectively preselectthis family {Hs,d(p)}? To answer this question, it isuseful to consider the following general decomposition ofHs,d:

Hs,d = Vsσ+d + V †s σ

−d + Vsσ

zd, (7)

where Vs and Vs are arbitrary system operators and ma-trices σ±d = 1

2 (σxd ± iσyd) act on the detector. In Eq. (7),

we discard any terms of the form ∼ Id, as those repre-sent the internal system evolution. Furthermore, for ourpurposes, it is also sufficient to consider a special case ofthe decomposition, where Vs = 0.

With Eq. (7) in mind, let us consider the transforma-tion of the system state ρs under a single cycle of thesteering protocol. First, let us consider the system statetransformation that is performed when the measurementoutcomes are averaged over (blind measurement). In theweak measurement limit, δt → 0, this is represented by

the map:

ρs → ΛVs(ρs)

≡(

1− δt2

2V †s Vs

)ρs

(1− δt2

2V †s Vs

)+ VsρsV

†s δt

2.

(8)

We note that the terms of order O(δt2) in this expres-sion represent the standard Lindbladian jump operator.Based on the map (8) by tracing out the detector read-outs after each step, one derives a Lindblad equation de-scribing the system evolution for the blind steering [18].

Let us now turn back to our protocol, where the dif-ferent measurement outcomes are discriminated. Duringstep 4 of the protocol cycle (cf. Definition 1), there is aprobability

p(cl) (ρs, Vs) = δt2 tr(VsρsV†s )

that a qubit flip is measured in the detector (click prob-ability). The resulting state in the limit of small δt isthen:

ρs → Λ(cl)Vs

(ρs) ≡VsρsV

†s

tr(VsρsV†s ). (9)

A “no-click” scenario occurs with probability

p(ncl) (ρs, Vs) = 1− p(cl) (ρs, Vs) ,

and results in a state:

ρs → Λ(ncl)Vs

(ρs) ≡

(1− δt2

2 V†s Vs

)ρs

(1− δt2

2 V†s Vs

)1− δt2tr(V †s Vsρs)

.

(10)Note that for the weak-measurement limit consideredhere (||Vsδt|| � 1), the click probability is parametri-cally smaller than that for the no-click event: a qubit flipcan be recorded in the detector only rarely.

2. Extensivity and locality

We are now in a position to expound our considera-tions for the family {Hs,d(p)} in terms of the operators{Vs(p)}. For a meaningful comparison between activeand passive protocols, we first require that there existsa passive protocol that employs Hamiltonians {Hs,d(p)}to reach the target state |ψ0〉. For concreteness, we as-sume that the passive protocol involving all the familymembers is a cyclic one: each of the couplings {Hs,d(p)}is employed one after another in a predefined manner,and the cycle is repeated once all the couplings are em-ployed. The size of the family is restricted by the numberof available system-detector couplings, which is assumedto scale up with increasing systems’ size not faster thanextensively.

It is then natural to demand that none of {Hs,d(p)}can move the system state away from the target state.

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Given Eqs. (9) and (10), this yields a dark-state conditionVs(p)|ψ0〉 = 0 for every p. This is equivalent to everyoperator Vs(p) taking the following form:

Vs =

D−1∑α=1

vα|ψ0〉〈ψα|+D−1∑α,β=1

wαβ |ψβ〉〈ψα|, (11)

where D is the Hilbert-space dimensionality of the sys-tem, and {|ψα〉} is any (many-body) basis for the systemthat includes |ψ0〉 as a basis state. This general formof the system-detector coupling is, however, not realisticfor many-body systems, as D = 2N grows exponentiallywith the number of qubits for an N -qubit system. Thus,having in mind steering of many-body states, we shouldfurther restrict the family of available steering operators.

The second condition for {Vs(p)} is that these cou-plings can realistically be engineered in an experimentalrealization of the system. In this work, we focus on themost basic aspect of this condition: locality. One mayconsider two types of locality: geometric and operator(k-locality [32]). Geometric locality of the operator Vsimplies that such interaction only requires coupling thesystem spins that are in geometrical proximity duringthe experiment. A k-local operator Vs implies that onlyk system spins are coupled at a time. It is natural toimpose the locality constraint not on the full operatorVs, but its individual terms. For example, if Vs involvesall system spins, but its individual terms only couple 2spins at a time, we will consider Vs a 2-local coupling (inline with [32]). Note that a k-local operator Vs impliesan interaction Hamiltonian Hs,d that is (k + 1)-local.

3. Sufficient conditions for the coupling operators.Room for active decision-making

It is worth stressing at this point that Vs(p) follow-ing the form given by Eq. (11) for all p is necessarybut not sufficient for |ψ0〉 to be the only dark state ofthe passive protocol. For some choices of such a fam-ily {Vs(p)}, a spurious final state |ψ′0〉 6= |ψ0〉 might bereached. However, this would imply a dark-state con-dition Vs(p)|ψ′0〉 = 0 (for every p), and this should nothold for a generic (say, random-matrix-type) operator Vs,which satisfies Vs|ψ0〉 = 0. For generic coefficients vα andωαβ in (11), one does not expect an existence of a spu-rious final state (for that, an extra constraint is needed,such as vanishing of certain vα, ωαβ , or a specific relationbetween the coefficients).

One concludes that a family consisting of a singleEq. (11)-type coupling Vs is sufficient to prepare |ψ0〉 ina passive protocol without generating a dark space. No-tably, reducing the family to a single member would leaveno room for active decision-making in a protocol definedby this family (an active protocol necessitates at leasttwo operators to choose from). On the other hand, suchan ultimate Vs would not generically satisfy the crucial

locality conditions and, thus, would be unrealistic to im-plement. Natural counterexample couplings V ′s that havemultiple dark states arise in the important case when V ′sacts only on a part of the system.

To construct such a counterexample, one may startfrom an arbitrary operator Vs that satisfies the dark-statecondition Vs |ψ0〉 = 0 for a single state |ψ0〉 in a givensystem. Now, consider a larger system embedding theoriginal one and construct a different target state whichis a tensor product of |ψ0〉 and a certain auxiliary state:

|Ψ0〉 ≡ |ψ0〉 ⊗ |ψ0〉. In this case, one may take Vs → V ′s ,where V ′s = Vs ⊗ Is still satisfies condition V ′s |Ψ0〉 = 0relative to this new target state in the extended Hilbertspace. Yet for a general starting state of the total system,the operator V ′s is obviously not sufficient to prepare theextended target state |Ψ0〉 – also implying the existenceof spurious dark states (in fact, all states of the form

|ψ0〉 ⊗ |ψ0〉 turn out to be dark, for arbitrary |ψ0〉).We see that in this “self-evident” fashion of selecting a

steering operator, the condition of |Ψ0〉 being a dark statefor a single operator V ′s was not sufficient for V ′s to beable to guarantee preparation |Ψ0〉. As discussed above,an operator Vs capable of steering a unique dark stateis typically highly nonlocal, in contrast to the limitedcapacity of a localized operator Vs ⊗ Is. We concludethat a family of multiple operators {Vs(p)} is neededto realistically prepare a target state, once that state issufficiently complicated. This, in turn, opens the doorfor gaining advantage through active decision-making.

III. TYPES OF SYSTEM-DETECTORCOUPLINGS

The preselected family of coupling operators {Vs(p)}determines both the performance of the ensuing passiveprotocols and the possibilities for active policy construc-tion. In the present section, we identify the crucial roleof the commutation properties of {Vs(p)}. We first con-sider N -qubit steering protocols which employ couplingoperators {Vs(p)} that are mutually commuting. As ashorthand, we denote this as non-frustrated steering. Weshow that a realistic passive protocol of this type canbe designed for product states and certain graph states.Commuting couplings also allow for a simple feedbackstrategy, which results in a significant speedup of therespective passive protocol. Next, we move on to pas-sive steering protocols that are frustrated. Such frus-tration of local couplings naturally arises for many-bodytarget states related to local parent Hamiltonians. Wepropose an explicit method of constructing a family ofnon-commuting operators {Vs(p)} that allows to preparesuch a many-body target state in a passive protocol. Thisforms the basis for Secs. IV and V, where we move on tothe active versions of frustrated steering protocols.

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A. Mutually commuting couplings

Here we focus on N -qubit steering protocols imple-mented with mutually commuting couplings {Vs(p)}. Aswill be demonstrated, a passive protocol of this typecan be constructed for an arbitrary target state, yield-ing an asymptotically precise passive preparation. How-ever, we find that this construction would, in general,require non-local system-detector couplings Hs,d, deem-ing the implementation of the protocol for many-bodystates impractical. We then identify an exception to thisrule – a subclass of graph states that can be obtained us-ing local commuting couplings. For this, we discuss theconstraints coming from both geometric locality, as wellas k-locality. Finally, we extend the discussion from suchpassive protocols to their active counterparts. To achievethis, we propose a simple feedback strategy that speedsup non-frustrated steering in a substantial way.

As a trivial example of non-frustrated steering, con-sider an N -qubit product state as a target state, e.g.,|00..0〉. The starting state will be assumed to be the per-fectly mixed state. To prepare it with a steering protocol,one can use a set of couplings parameterized by the qubitnumber i = 1, ..N :

V (prod)s (i) = γσ−i . (12)

Passively alternating between the steering cycles employ-

ing V(prod)s (i) with different i guarantees preparation of

the target state with any given accuracy. This directlyfollows from the analysis of Secs. II B and II C. For an ac-tive version of the protocol, partial protocol terminationcan be applied: if a click is registered when measuring

any qubit i, the coupling V(prod)s (i) is dropped out from

the sequence of couplings that will be applied in furthercycles. In other words, the steering with this “fired”coupling is terminated at this point, whereas other cou-plings remain active – hence the term “partial termina-tion”. Since this implies a readout-based decision on theset of steering couplings that are used at a given step,we classify this as an active steering protocol. In theε→ 0 limit, this strategy results in the following relationbetween active and passive runtimes:

N (act)c (ε) =

N(pass)c (ε)

2+

N

2γ2δt2, (13)

which leads to up to a substantial 2-fold speedup for theactive version, similarly to Eq. (6).

Non-frustrated steering towards any target state |ψ0〉can in principle be designed if we allow for an arbi-trary coupling set. Indeed, given a many-body unitarytransformation to |ψ0〉 from a product state |00..0〉, i.e.,|ψ0〉 = Uψ |00..0〉, one may formally construct a family ofcouplings:

V(Uψ)s (i) = γUψσ

−i U†ψ. (14)

Clearly, any protocol for |ψ0〉 preparation using couplingsof the form of Eq. (14) would be a unitary equivalent of

the same protocol which uses couplings of Eq. (12) toprepare |00..0〉. Therefore, a passive protocol iterating

over V(Uψ)s (i) for different i would successfully prepare

the target state |ψ0〉. We also conclude that a partial-termination strategy can be applied to this coupling setwith the same effect as for the product state target. Note,

however, that in most cases employing V(Uψ)s (i) would

not be practically feasible. Indeed, since Uψ is a general

many-body operation, the couplings V(Uψ)s (i) would in-

volve arbitrarily non-local terms. For most N -spin states|ψ0〉 with large N , one thus expects that the resulting

V(Uψ)s (i) would break any requirement of geometric ork-locality.

This locality-violation rule can be circumvented for Uψwhich is given by a shallow circuit and thus |ψ0〉 which isweakly entangled. As a resourceful example of such |ψ0〉,consider a graph state defined on a generic graph G [33]:

|ψG〉 =

∏(j,k)∈

edges(G)

U(gr)(j,k)

( |0〉+ |1〉√2

)⊗N, (15)

U(gr)(a,b) = exp (iπ|00〉〈00|a,b) , (16)

in which case

Uψ =

∏(j,k)∈

edges(G)

U(gr)(j,k)

∏j∈qubits

exp(

iπ

4σyj

) .

Since two-qubit rotations U(gr)(j,k) all mutually commute,

the coupling V(Uψ)s (i) acts only on spin i and on the

spins j whose vertices share an edge with i in the graphG. Therefore, this coupling is (k + 1)-local if there are

k edges coming out of vertex i. Moreover, V(Uψ)s (i) is

also geometrically local, if the graph G only connectsthe qubits which are in geometric proximity. We con-clude that for the graphs satisfying the above conditions,a realistic preparation of graph states with local non-frustrated steering is possible. Such a protocol can besped up in the same way it was possible for the productstates – using active feedback via the partial-terminationstrategy.

For the perfectly mixed starting state, the partial-termination policy gives an optimal speed-up of a pro-

tocol driven by non-frustrated couplings V(Uψ)s (i). In-

deed, the protocols in question are then equivalent toan independent set of N 1-qubit steering protocols (un-der the unitary transformation Uψ). This picture, how-ever, breaks down for a more general starting state.Let us first consider the trivial target Uψ = I, |ψ0〉 =|00..0〉, while the starting state is itself entangled (e.g.1√2(|00..0〉 + |11..1〉)). In this case, the click received

from a single coupling Vs(i) may imply that multiple cou-plings can be dropped from the applied sequence, and not

8

just Vs(i) itself. This would be more optimal than thepartial termination strategy outlined above. The samepicture extends to the more interesting case when thetarget state |ψ0〉 is entangled itself, e.g. a graph state,while the starting state is a product state. Indeed, un-der the unitary mapping Uψ which takes an entangledstate |ψ0〉 to |00..0〉, the product starting state in turnbecomes entangled. Hence, the previous reasoning ap-plies and partial-termination would generally not be anoptimal active policy in this situation. Instead, one mayaccelerate it further by applying one of the frustrated-coupling strategies outlined in the following sections.

B. Frustrated system-detector couplings

From now on, let us focus on accelerating steering pro-tocols which employ couplings constrained by at leastone of the two notions of locality. Under this premise,for target states other than the product states and statesprepared by a shallow circuit, we would generally need togo beyond the non-frustrated protocols outlined above.The first question to tackle is how to design the localcouplings Vs that are suitable for a passive protocol. Inprinciple, this can be addressed on a case-by-case basis,tailoring some coupling set with a specific target statein mind. (This approach will be demonstrated for theW -state preparation in Sec. V D.) However, this is notalways a straightforward task. Therefore, it is interest-ing to know whether one can devise a general scheme tothis end. For this, we propose an approach based on aparent-Hamiltonian construction.

The parent Hamiltonian Hψ of |ψ0〉 has |ψ0〉 as a non-degenerate ground state, and is constructed from |ψ0〉 inthe form:

Hψ =∑j

H(j)ψ . (17)

Here, all the terms H(j)ψ , while defined as acting on the

entire Hilbert space, are local in the real space given thatthe state |ψ0〉 hosts a limited amount of entanglement[34] (implying an area-law dependence of the entangle-ment entropy accommodated in |ψ0〉). Note that, by the

construction of the parent Hamiltonian, terms H(j)ψ have

|ψ0〉 as their common ground state, although they gener-ally do not commute with each other: this is possible be-cause of their respective ground state degeneracy. These

degenerate ground spaces of H(j)ψ will be the central el-

ement of our construction of the coupling family. For

the term H(j)ψ , which nontrivially acts on some m qubits,

let us denote its m-qubit ground states as |φ(j)a 〉 and the

excited states |θ(j)a 〉. Given these, we can construct the

coupling operators of the following form:

V js (w,v,u) =∑ab

wab |φ(j)a 〉 〈θ

(j)b |

+∑ab

vab |θ(j)a 〉 〈θ

(j)b |+

∑ab

uab |φ(j)a 〉 〈φ

(j)b | . (18)

A particular example of this construction will be ad-dressed in detail in the context of the AKLT model inSec. IV (see also Refs. [11, 18]).

For a generic (fixed) value of the parameters involved,running a passive protocol with the coupling given byEq. (18) allows one to steer the system into the ground

state of H(j)ψ . Alternating the coupling operators by se-

lecting terms with different j at different measurementsteps allows for steering the system into the joint ground

space of all couplings H(j)ψ . This space is given by the

target state |ψ0〉 only, as it is the non-degenerate groundstate of Hψ. Thereby, as long as the parent HamiltonianHψ is local, we have managed to construct an appropri-ate coupling set for a passive protocol (also see [11] for arelated statement proven in more detail).

Now, let us consider an active-protocol construction.First, we note that unlike in the “non-frustrated” proto-col construction, the operators V js for different choices ofparameter sets are, in general, not mutually commuting.This also applies to the couplings with different values

of j, as H(j)ψ generally do not commute. Therefore, the

measurement outcome of steering by V js (where, because

of the locality of H(j)ψ , j corresponds to a certain region

in real space) impacts the outcomes of steering at otherlocations. As a result, the partial-termination strategycannot be applied to this coupling set, as it assumes thatthe respective cycles of the protocol can be consideredseparately. Instead, the feedback strategy for the frus-trated steering should continuously coordinate the appli-cation of different couplings in the protocol. In a many-body context, this becomes a complicated navigation-type problem (cf. Ref. [22]). We devote the followingtwo sections to the study of such possible coordinationpolicies.

IV. QUANTUM COMPASS: COST-FUNCTIONPOLICIES

One way to enable the Hilbert-space navigation is tointroduce a cost function C(ρs), which is to be minimizedin the protocol. The basic example would be the infi-delity C(ρs) = R(ρs, |ψ0〉) of the system state ρs to thetarget state |ψ0〉, defined in Eq. (1). Achieving the globalminimum R(ρs, |ψ0〉) = 0 of this cost function would beequivalent to preparing the target state. In general, tocalculate R(ρs, |ψ0〉), one needs to know the state of thesystem ρs. This is, in principle, feasible, as we controlthe system evolution given all measurements outcomesand therefore can numerically simulate it in parallel tothe experiment.

9

4 6N

0

100

200

300

400

Meanruntim

es

4 6N

4

6

8

10

Speedu

pfactor

Performance scaling with system size

0 50 100 150 200Applied protocol cycles

Total runtime

Infid

elity

R

10 2

10 1

100Preparing AKLT state: trajectories(a)

(b)

(c)

Preparing AKLT state: statistics

Passive protocolActive protocol

—

—

—

—

0 200 400 600

10 4

10 3

10 2

10 1

Count

fraction

Passive protocolActive protocol

FIG. 2. (a) Fidelity as a function of the protocol cyclefor active and passive protocol runs towards a 5-spin AKLTstate. These example runs are characterized by the durationsimilar to the mean protocol durations of respective proto-cols (≈ 199 ± 4 for passive and ≈ 28 ± 0.5 for active proto-col) (b) Histograms of protocol durations t for the prepara-tion of the five-spin AKLT, with accuracy given by infidelityR < ε = 0.01. An exponential decaying profile, characteris-tic of a Poissonian process, can be clearly observed (note thelog scale). Note that all recorded runs for an active protocollasted far less than the mean duration of a passive protocol(200 cycles). Each histogram was compiled from 104 simu-lated runs; the figure is truncated at 600 cycles for betterpresentation. (c) Scaling of the active protocol’s advantagewith system size N . A speedup factor tends to increase sig-nificantly as the system scales, with factor 9.5 being the es-timated speedup at 6 spins. The error bars represent 95%confidence intervals due to sampling error in numerical simu-lation. 104 samples were collected to simulate both protocolsN = 3, 4, 5, and 103 at N = 6. Similarly to the above, theinfidelity threshold is ε = 0.01.

However, the requirement of such a simulation being donein parallel to the experiment puts a restriction on thesize of the system that one can work with. For now, wewill accept this limitation; finding ways to mitigate it isamong the worthwhile potential extensions of this work.

With a given cost function C(ρs) at hand, we canuse it to form the active decision for the coupling op-erator Vs(p). The ultimate strategy is to pick Vs(p)which brings the system to the global minimum |ψ0〉in the fastest expected time. For C(ρs) = R(ρs, |ψ0〉)this is equivalent to the ultimate strategy defined by dy-namic programming [29], requiring unrealistic computa-tion power. Therefore, the notion of the global cost func-tion does not give any additional advantage in construct-ing such a strategy. Instead, one can use its cheaperversion – the “greedy strategy.” Specifically, one can useVs(p) that yields the fastest expected reduction of thecost function in a single step of the evolution:

V (greed)s (p) = argminVs(p)R[ΛVs(p)(ρs)], (19)

where ΛVs(p)(ρs) is defined in Eq. (8) If there are multipleminima, we will assume that argmin returns a randomrepresentative among those. With only a small amountof computations needed to decide for the optimal next

coupling V(greed)s (p), this greedy procedure allows us to

avoid the complex long-term analysis of the protocol.As one can see from a direct implementation, the

greedy minimization of the cost function can acceleratethe state preparation by a large factor. To demonstratethis, we consider the example of the ground state in anAKLT spin chain as the target state. This is an entangledstate of N spin-1 particles governed by the Hamiltonian:

HAKLT =∑i

Hi,i+1 =∑i

[~Si · ~Si+1 +

1

3

(~Si · ~Si+1

)2],

(20)Here, we assume periodic boundary conditions, implyinga single ground state [30]. The Hamiltonian (20) is aparent Hamiltonian (as defined in Sec. III B), where eachterm Hi,i+1 has four degenerate ground states |φia〉 witheigenvalue −2/3, and 5 excited states |θib〉 with energy4/3.

Hereafter, we consider the all-down product state asour starting state. We first design a passive steering pro-tocol for AKLT state preparation, following the parentHamiltonian construction from Sec. III B. For simplicity,we restrict ourselves to a less general version of Eq. (18),and use the following family of coupling operators (cf.Refs. [11, 18]):

V (c, i) = |φi4〉 〈θi5|+∑

α,β=1,..4

cαβ |φiα〉 〈θiβ | , (21)

with cαβ constrained to be an orthogonal matrix, to makesure that the interaction is of constant strength and thusno bias is introduced in the construction. In a passivesteering protocol, we will alternate between different val-ues of i, while drawing instances of orthogonal matrices

10

c at random. For an active feedback strategy to be usedon top of this, we propose a greedy policy relative toC(ρs) = R(ρs, |ψ0〉) to select c. In both passive andactive protocol, we assume each coupling to be appliedmultiple times until one either receives a click, or no-click for an asymptotically long time. Such a repeatedapplication of a single coupling is then counted as a sin-gle protocol cycle. We take this approach for a practicalpurpose because simulating such protocols is more acces-sible numerically.

Thus simulated, the relative performance of the passiveand the active policies (Fig. 2) shows a strong advantageof the active policy. In particular, the speedup factor issteadily increasing with system size (Fig. 2c), reachingthe value of 9.5 for N = 6.

A. Discussion: orthogonality catastrophe andalternative cost functions

The approach defined above harbors a potential chal-lenge. For the greedy procedure to be effective, itshould always yield a nonzero bias in favor of a specific

V(greed)s (p) (or a small subset thereof). In other words,

the landscape of the cost function C(ρs) should not beflat — and some cost functions may yield better land-scapes than others. In particular, applying the infidelitymeasure R(ρs, |ψ0〉) is, in general, fundamentally flawed.Indeed, a (2N − 1)-dimensional subspace of states in theN -body Hilbert space is orthogonal to the target state.Let us consider the case when the starting state belongsto that subspace. This situation would in general notchange after a single steering cycle with a local couplingVs(p). For our purposes, it implies that the infidelitymeasure R is equal to 1 for a large manifold of states,and there might be no direction of increase that wouldallow us to choose an appropriate coupling. The most di-rect example of this can be observed when applying thegreedy policy to non-frustrated steering (see Sec. III A).For simplicity, let us again take the product state of Nqubits |00..0〉 as the target state, the state |11..1〉 as thestarting state, and the couplings V (i) = σ−i for steering.Only after such steering protocol results in N successfulclick events, R(ρs, |ψ0〉) gains a nonzero value. Thus be-fore N − 1 clicks, the greedy policy for R(inf) will notbe capable of providing a meaningful decision for thenext coupling. Strongly enhanced by the system size,this phenomenon is reminiscent of Anderson’s orthogo-nality catastrophe [35].

As a remedy to this deficiency, a “subsystem infidelity”measure can be introduced:

RS(ρs, |ψ0〉) =∑σ∈S

[1− tr

(√√ρ0,σρs,σ

√ρ0,σ

)2],

(22)where ρ0,σ (ρs,σ) is the reduced density matrix of the tar-get state (current state) with respect to subsystem σ. Sis the family of subsystems from which σ are drawn; the

choice of S depends per target state. In the case of the|11..1〉 → |00..0〉 protocol described above, the appropri-ate S would be the set of individual spins. Unlike R, suchquantity RS changes every time when a click occurs inthis protocol. As a result, the greedy policy with respectto the local RS would yield the partial-termination pro-tocol of Sec. III A, significantly boosting the preparationof such a product state.

By continuity with the case of the product state tar-get, such preference for RS should extend to the weakly-entangled target states, and maybe to some highly-entangled targets. However, we did not see a manifes-tation of this in the case of our AKLT simulation, whereusing RS as a cost function did not yield any improve-ment compared to R. As a likely explanation for this, theorthogonality catastrophe should become manifest onlyat large system sizes, where the classical simulation ofthe protocol is also hindered. However, we expect thatsome practical target states may still develop a noticeableperformance difference between RS and R, similarly tothe case of the product state target. A further study ofthis question constitutes a promising direction for futurework.

V. HILBERT-SPACE ORIENTEERING MAP:QUANTUM STATE MACHINE

In this section, we present an orienteering tool that isan alternative to cost-function minimization: mappingout the steering transformations with a Quantum StateMachine (QSM) construction. We then illustrate naviga-tion in many-body Hilbert space, employing this machin-ery to the preparation of the highly entangled W-state ofthree qubits.

A. QSM generalities

Every transformation of the system’s state, Λ(cl)Vs(p) and

Λ(nc)Vs(p), associated to steering with a specific coupling

Vs(p) in a given readout scenario [click or no-click, re-spectively, see Eqs. (9) and (10)], can be represented witha directed graph with complex weights. For this, we no-tice that every such steering transformation conserves thepurity of the state. Therefore, it is convenient to encode

transformations Λ(cl, ncl)Vs

in their action on Hilbert spacebasis states |φα〉:

Λ(cl,ncl)Vs

(|φα〉) =1√

p(cl,ncl)

∑β

L(cl,ncl)αβ |φβ〉 (23)

L(cl)αβ = 〈φβ | δtVs |φα〉 , (24)

L(ncl)αβ = 〈φβ | 1− δt2V †s Vs/2 |φα〉 , (25)

where p(cl) (p(ncl)) is the probability of a click (non-click)readout upon this steering action. Note that in Eq. (23),

11

we extended the action of ΛVs to pure states by a slightabuse of notation compared to Eq. (9).

The graph representation for steering action Λ(cl,ncl)Vs(p) ,

or a steering graph, is then directly obtained from the

amplitudes L(cl,ncl)αβ . The vertices in such a graph corre-

spond to the Hilbert space basis states, and the edgesdescribe the steering transformations. The edges are di-rected and weighted with complex amplitudes, the edge

α → β being weighted with amplitude L(cl, ncl)αβ (edges

weighted with zero amplitudes are excluded from thegraph). Implying this definition, we will use the notationL(cl, ncl) for the steering graphs themselves. For basicexamples of steering graphs, please refer to Fig. 3.

Since the weights L(cl)αβ are proportional to the matrix

elements of coupling operator Vs while L(ncl)αβ can be ex-

pressed via Vs as well, the graph L(ncl) for the no-clickaction can be inferred entirely from the graph L(cl) forthe click action. In particular, due to the term ∝ V †s Vs,

graph L(ncl) contains an edge e(ncl)ij from vertex vi to vj ,

if a graph L(cl) contains edges e(cl)ik and e

(cl)jk (see Fig. 3).

Heuristically, to yield a L(ncl)-edge, one has to first followa L(cl)-edge forward, and then another L(cl)-edge back-ward. Furthermore, due to the additional identity oper-ator term in Eq. (25), any graph for the no-click steeringaction will also include self-loops on each vertex.

The steering graphs introduced above can now be usedto create a Quantum State Machine. For nV couplingsVs(p) in the steering kit, there exist 2nV graphs corre-

sponding to steering maps Λ(cl, ncl)Vs(p) , because of the two

possible measurement outcomes for each of the couplings.The QSM for the steering protocol is then obtained as acollection of these graphs. It can be represented as acolored multigraph, where each steering graph is repre-sented as a single-color subgraph (Fig. 4). Consequently,in a QSM multigraph there may be multiple edges goingfrom any vertex α into any other vertex β (making it amultigraph rather than a simple graph), but at most onesuch edge for each color.

Let us now consider our original task of finding the ac-celerated navigation protocol. To make use of the QSMconstruction in this context, we will restrict our consid-eration to bases {|φβ〉} where one of the basis states isthe target state |ψ0〉 itself. In such a case, state |ψ0〉corresponds to a marked vertex in the graph, and thegoal of the steering protocol becomes to drive the systemstate to that vertex. The goal of optimizing this protocolmay then look similar to a known problem of finding theshortest path to the marked vertex on a weighted graph.This problem is standard in graph theory and can besolved as such. Can such a solution be used to designthe navigation protocol?

As we will see in Sec. V B, this analogy is not com-plete, since the quantum evolution on the graph goes be-yond the simple path-on-the-graph picture. This aspectcreates an obstacle to directly applying the graph ex-

0 0

2 1

1

FIG. 3. Examples of steering graphs (see definition inSec. V A): (a) Steering graphs on a 3-level system, corre-

sponding to the coupling Vs = γ(|1〉 〈0|+ |1〉 〈2|). Graph L(cl)

for click action is denoted with solid arrows and the graphL(ncl) for no-click action by dashed arrows (a particular rep-resentation of graph coloring). Note that due to the identityoperator in Eq. (25), every vertex is decorated with a self-

loop from the L(ncl) graph. To see how the rest of L(ncl) canbe deduced from L(cl) (cf. discussion in Sec. V A), consider

the example of e(ncl)02 (dashed arrow from state 0 to 2). Ac-

cording to the graphical approach from Sec. V A, one is to

follow edge e(cl)01 (solid arrow from 0 to 1) forward and then

e(cl)21 (solid arrow from 2 to 1) backward - and thus manages

to travel from state 0 to 2, in correspondence to e(ncl)02 . (b)

Steering graphs on a 2-level system, as defined by the cou-pling Vs = γ(|1〉 〈1| + |1〉 〈0|). Following the same rule as

above, inter-vertex edges of L(ncl) can be deduced from L(cl).

For example, by following the edge e(cl)11 forward and then the

edge e(cl)01 backward, one performs a transition from state 1 to

state 0, thus reproducing the edge e(ncl)10 from L(ncl).

ploration algorithms to facilitate our protocol speed-up.Fortunately, in some cases, this difficulty can be properlyaccounted for, as we will see in Sec. V C. In those cases,the “semi-classical heuristics” of graph exploration mayindeed be applied. Finally, in Sec. V D, we will apply thisapproach to the preparation of the W -state, with a fac-tor 12.5 improvement compared to the passive protocolusing the active navigation protocol.

B. Quantum subgraphs in a QSM

Let us now compare our QSM navigation task to thestandard problem of graph exploration. Our goal is toidentify the differences between the two, which prevent usfrom applying the graph exploration techniques directlyto QSM navigation. First of all, the state of the system ingraph exploration is at all times represented by a singlevertex. The system in a QSM, on the other hand, isgenerally represented by a superposition over multiplevertices. Furthermore, in graph exploration, the stateis modified by following one of the edges. A steeringaction in a QSM, in contrast, corresponds to a wholecollection of edges – i.e., a single steering graph in theQSM multigraph.

Some steering graphs may induce quantum effects,

12

0 1 2

FIG. 4. A basic example of the QSM multigraph, describingthe steering kit for a three-state system. The steering op-tions are represented by the coupling operators V1 = γ1 |1〉 〈0|and V2 = γ2 |2〉 〈1|. The starting state is 0, marked in blue,and the target state is 2, marked in green. The optimal co-ordination policy of the two steering operations is straight-forward: one needs to first repeatedly apply the V1-steeringuntil a click is obtained, and then the V2-steering until a clickis obtained. Compared to the passive steering which iteratesbetween V1 and V2 regardless of measurement outcomes, thisdirectly yields a 2-fold speedup in the average performance.

such as superposition and interference. For instance,the steering action whose graph contains two outgoing

edges from a given vertex (e.g., vertex 0 for graph L(cl)1

in Fig. 5a), can create a nontrivial quantum superposi-tion. If a state is given by a superposition of multiplevertex states, it may further undergo quantum interfer-ence. In particular, this can be facilitated by a steeringaction whose graph contains a vertex with two incom-

ing edges (e.g., vertex 4 for graph L(cl)2 in Fig. 5a). In

general, a notion of “superposition subgraphs” and “in-terference subgraphs” of a steering graph can be defined:

1. Superposition subgraph is a subgraph of a steer-ing graph span by multiple (more than one) edgesoutgoing from a single vertex.

2. Interference subgraph is a subgraph of a steeringgraph span by multiple edges incoming to a singlevertex.

Collectively, we will refer to such interference and su-perposition subgraphs of a single steering graph as itsquantum subgraphs. If the quantum subgraphs are ab-sent in the QSM, we will refer to it as a classical QSM.In other words, in a classical QSM, each vertex has atmost one outgoing and at most one incoming edge of anygiven color.

If a QSM is classical, optimization of the navigationprotocol can essentially be reduced to classical graph ex-ploration. For a simple example of a classical QSM andthe way to optimize the respective state preparation, con-sider the 3-level steering actions described in Fig. 4. Notethat optimization of the classical QSM also applies to thecase when the starting state is a superposition of multiplevertex states. If the steering operations contain no quan-tum subgraphs, the quantum superposition is equivalentto a probabilistic mixture for the sake of the protocol

0

1

2

3

4

+

-

0

3

4

FIG. 5. Possible configurations of quantum subgraphs ina QSM, exemplified by the 5-vertex subgraph of an exam-ple QSM. (a) The click-action graphs for the three couplingoperators V1,2,3 that form the steering kit. The operatorshave the form V1 = γ1(|1〉 − |2〉) 〈0|, V2 = γ2 |4〉 (〈1| − 〈2|),V3 = γ3 |3〉 〈1|. The graphs for the no-click actions are notshown, as their form can be deduced from the graphs for clickactions. In the present basis, the V1-click is manifest as asuperposition, the V2-click – as an interference, and the V3-click corresponds to a semiclassical evolution. (b) QuantumState Machine for the steering kit from the previous panel,depicted in a different basis. The basis transformation is|±〉 = (|1〉 ± |2〉)/

√2. In this case, the basis transformation

removes the quantum elements in the L(cl)1,2 graphs, however,

it turns L(cl)3 into an interference element. Note that there is

no basis transformation that would turn such a QSM into aclassical one. This statement follows from the uniqueness ofthe Jordan canonical form for operators V2 and V3.

optimization, and the optimal navigation pattern can beextracted accordingly.

As the form of the steering graph depends on the choiceof basis, it is conceivable that the number of quantumsubgraphs in such a graph in some cases can be reducedby changing the basis (compare Fig. 5a and b). However,using a change of basis to remove all the quantum sub-graphs in an arbitrary QSM is generally impossible (seeFig. 5).

C. Coarse-grained QSM. Semiclassical heuristic fornavigation

We now focus on the steering protocols whose QSMcannot be made classical via a basis transformation. Insuch a case, it may still be possible to optimize it via aclassical graph exploration heuristic. For that, we pro-pose to coarse-grain the QSM by grouping subsets of itsvertices into single block-vertices. The coarse-grainedQSM would consist of graphs drawn between such block-vertices. The block-vertex containing the target vertexcan be considered as the target block-vertex.

An inter-block edge between two block-vertices isdrawn, if the original QSM has at least one edge con-necting the vertices inside the respective block-vertices.For the coarse-graining to be useful for our purposes,it should be done in such a way that all of the result-ing QSM graphs have a classical structure. Namely,

13

the coarse-grained graph should not have quantum sub-graphs, e.g. realizing superposition or interference be-tween the block-vertices (in analogy to Sec. V B). To sat-isfy this requirement, the following rule for vertex group-ing can be employed (cf. Fig. 6): if two edges of thesame color are simultaneously coming in or out of a givenvertex, the two vertices at the other ends of these edgesshould be grouped within one effective block-vertex. Thisrule manifestly yields basis-dependent groupings, sincethe very presence of quantum subgraphs in a QSM isbasis-dependent. Thus, a smart choice of the basis mayallow for an efficient and simpler coarse-grained graph.Designing a general explicit algorithm for finding theoptimum basis for an arbitrary QSM is a highly non-trivial task. Heuristically speaking, a convenient choiceof the basis should be the one that results in the min-imum number of quantum subgraphs in a QSM beforecoarse-graining.

1

02

3

4

FIG. 6. Semiclassical coarse-graining applied to a QSM.(a) A 5-state part of a QSM with two quantum subgraphs:

interference subgraph realized by L(cl)2 and a superposition

subgraph realized by L(cl)1 . Since pairs of states {|0〉 , |1〉}

and {|3〉 , |4〉} fall under conditions described in Sec. V C,these are to be grouped together in a coarse-grained QSM.(b) Simplified depiction of a coarse-grained QSM, obtainedfrom (a).

For the coarse-grained graph to be effectively classi-cal, we desire to ignore details of the system evolutioninside the subspace of a given block-vertex. Specifically,we aim to view every block-vertex as an effective sin-gle state of the system and assume that every edge al-lows transporting the system between such block-vertexstates with no obstacles. If this was directly possible,and since the coarse-grained QSM by definition containsno quantum subgraphs, optimization of its explorationwould have become a classical task. However, such an

approximation scheme needs more careful justification.Every block fundamentally corresponds to a Hilbert sub-space, and an inter-block edge is given by a N1 × N2

matrix of coefficients (where N1 and N2 are the inter-nal dimensionalities of the linked blocks). Characteriz-ing these effectively with single amplitudes may lead toerroneous navigation policies. In particular, one state in-ternal to a block-vertex might be untouched by an inter-block edge, i.e., it only yields zero matrix elements in amatrix characterizing the edge. If the edge is outgoing, asystem initialized in the said state would not be able toescape the block-vertex using that edge alone (see Fig. 7).This is in direct conflict with characterizing blocks andinter-block edges with single amplitudes. For an incom-ing edge, a similar problem may arise: some states insidea block-vertex might not get populated when that edge isactivated. This may become detrimental for the naviga-tion protocol based on a coarse-grained QSM, especiallyif the unavailable state in question is the final target ofthe protocol.

Such difficulties may be overcome, if some of the cou-plings given in a QSM allow for an internal mixing of thesubspace (represented by a self-loop on the block-vertexin the respective L(cl)-graph). Applying such a couplingin the protocol would allow one to make the block-vertexaccessible to all the edges that are connected to it (seeFig. 7), via a sufficient number of clicks. In the sce-narios described above, where additional couplings areneeded to turn a block-vertex into an effective singlevertex, we will refer to such couplings as ancillary cou-plings. Note that given a steering kit, there is no guaran-tee that the ancillary couplings needed for exploration ofevery block-vertex, are available. For simplicity, in thiswork, we restrict our further consideration to the coarse-grained QSMs, where the ancillary couplings happen tobe present wherever needed. Every block-vertex can thenbe made accessible to the outgoing edges, and the targetstate is ensured to be reachable once the target blockis reached. In this case, we consider the coarse-grainedQSM as effectively semiclassical.

To design an active steering policy within the coarse-grained approach, we note that the navigation proto-col has the following structure. The system state canbe transported between block-vertices, and eventuallysteered to the target block-vertex. After that, either thetarget state is reached already (one can obtain this in-formation from the simulated copy of the system), or itcan be reached after applying ancillary couplings on thetarget block-vertex. The cost of the protocol can nowbe broken into two parts. The first is the cost of explor-ing the coarse-grained graph using the inter-vertex edges.The second is the dwell time inside the block-vertices,which is spent applying the ancillary couplings. If wecould find the route through the graph that minimizesthe combination of these two components, it would solveour optimization problem exactly. However, because ofthe presence of the degrees of freedom that are internalto the block-vertices, the coarse-grained geometrical in-

14

0

3

12

FIG. 7. Illustration of ancillary couplings in the contextof QSM coarse-graining (a) A 4-state part of a QSM thatis subject to coarse-graining, featuring non-trivial actions bycouplings denoted as V1 and V2. States |1〉 and |2〉 are tobe grouped together since they are both targets in a super-position subgraph span by edges e02 and e01 of a steering

graph L(cl)1 . (b) The coarse-grained version of the QSM from

(a). The block {|1〉 , |2〉} is connected to state |3〉 through

an outgoing edge of L(cl)1 . However, from microscopic point

of view exemplified in (a), no population can be transferred

from state |2〉 to |3〉 unless the click action Λ(cl)2 is realized

first. Therefore, including and applying V2 as an ancillarycoupling is required for a valid semiclassical coarse-grainingof this QSM.

formation does not allow for such a precise solution. Inother words, both the inter-vertex travel time and theblock-vertex dwell time depend on the microscopic de-tails of the evolution.

Instead of studying such quantum-mechanical micro-scopics, we propose a semiclassical approximation to thiscalculation. Specifically, we assign every inter-block edgea characteristic traversal time, and every block-vertex acharacteristic dwell time. For this, we use the matricesfor click transitions between blocks i and j (the case ofancillary couplings given by i = j). Let us loosely denote

these as L(cl)i,α;j,β , implying that only matrix elements with

states from blocks i and j are included. In that case, theeffective transition amplitude between blocks i and j can

be defined as operator norm L(cl)i,j = ‖L(cl)

i,α;j,β‖, and char-

acteristic traversal (dwell if i = j) time τi,j = (L(cl)i,j )−2.

Note that this reduces to the average traversal time forthe case of a genuinely classical graph, with an amplitudeγδt connecting two states implying duration of τ = 1

γ2δt2

for traversal (cf. Sec. II B).With characteristic times τi,j assigned, the time-cost of

following a specific path through this graph can be esti-mated as a combined characteristic time of all the edgesand vertices crossed along the way. The desired pathwill be the one that optimizes this expected time. Onthe one hand, this may result in a different navigationprotocol compared to what is optimal from the completequantum-mechanical analysis. On the other hand, sucha first-principles analysis is prohibitively hard, and weexpect that our semiclassically derived protocol will stillbe considerably quicker than its completely passive ver-sion. One example of such an improved protocol is givenbelow.

D. W-state preparation

To illustrate the principles of the QSM framework, weconsider the coarse-graining approach to the navigationof a 3-qubit state from a trivial |000〉 state to a so-calledW-state [36] that has the following form:

W =1√3

(|100〉+ |010〉+ |001〉). (26)

For the steering kit, we choose the following family ofcouplings (assuming labels A, B, and C for the qubits):

V1 = σ+A − σ

+C , (27)

V2 = σ−Aσ−C , (28)

V3 = σ−Aσ+B − P

0AP

1B , (29)

V4 = σ+Bσ−C − P

1BP

0C . (30)

Here, σ± = 12 (σx ± iσy) and P a = |a〉 〈a| , a = 0, 1. A

passive version of the protocol would amount to blindlyalternating between the steering actions with different Vi,which does yield the target state, given that the steeringis applied a sufficient number of times (see Fig. 8.b).

To design a feedback policy, we now consider a QSMrepresentation of the steering kit. It is shown in Fig. 8a.Note that this QSM has multiple quantum subgraphs.Therefore, to employ a feedback policy, it should be sub-jected to the subspace-clustering coarse-graining tech-nique, as outlined in Sec. V C. It proves useful to clus-ter the Hilbert space by the total excitation number,which results in a semiclassical QSM, as desired (Fig.8b). Given the starting state of the evolution, it is thenstraightforward to design the policy that leads to the tar-get state:

1. Repeat V1-steering until a click is obtained;

2. Repeat V3-steering until the target state is reached(with fidelity error below ε).

This protocol moves the state of the system from thezero excitation state to the single-excitation subspace(part 1) and then takes the system to the W-state in

15

FIG. 8. Measurement-driven navigation towards the 3-qubitW -state: QSM representation. (a) Steering with couplingsEqs. (27)-(30). The vertices in the single-excitation sub-space are given by states |W 〉, |φ−〉 ≡ 1√

2(|100〉 − |001〉), and

|φ+−〉 ≡ 1√6(|100〉 − 2 |010〉 + |001〉). (b) The coarse-grained

version of the above QSM. The vertices are labeled by theexcitation number. From perspective of Sec. V C, couplings2 and 3 play the ancillary role. Indeed, those couplings mixthe internal structure of the block-vertices, allowing one toeventually steer the state to the target |W 〉.

that subspace (part 2). Note that this employs only twocouplings out of four, a simplification that is only pos-sible with an active steering protocol. In a passive pro-tocol, as all couplings are employed cyclically, multipleV1-clicks may accidentally occur before the target stateis reached (Fig. 9a), and, therefore, other couplings areneeded to reduce the excitation number back to 1. InFig. 9b, the performance histogram is given for a largenumber of numerical trials for the active and passive pro-tocols. Both are run with δt = 0.1 for the target fidelityerror ε = 0.01. We are primarily interested in the av-erage runtimes of the active and the passive protocol.These are Nact ' 365 ± 3 and Npas ' 4600 ± 100 cyclesof the protocol, respectively, yielding a speedup factor ofaround 12.5.

VI. DISCUSSION AND CONCLUSIONS

In this work, we have put forward the concept ofmeasurement-driven active-decision steering of quantumstates. We have developed steering protocols in whichthe measurement readouts are used to adjust the mea-surement protocol on-the-go, yielding significant acceler-ation of state preparation, with improved fidelity, com-pared with passive steering. The possibility of exploit-ing the readouts explored here is the great advantageof measurement-based steering over drive-and-dissipation

Total runtime0 150005000 10000

10 5

10 4

10 3

Cou

ntfra

ctio

n Passive protocolActive protocol

0 2000 40000

1

2

3

Active protocolPassive protocolSu

bspa

cenu

mbe

r

(a)

(b)

Preparing W-state: trajectories

Preparing W-state: success statistics

Applied protocol cycles

—

—

—

FIG. 9. Performance of the active navigation protocol basedon the QSM representation of steering towards the W -state,as described in the text. (a) Typical trajectories of the pas-sive and active protocols, in terms of the excitation numbersector that is occupied by the system state. Displayed aretrajectories that yield the runtimes approximately equal toaverage runtimes of 365 (active) and 4600 (passive). The firstV1-click in the displayed run of the active protocol occurs asearly as the 14th cycle, which is not legible from the plot.(b) The histogram over the protocol runtimes, required fora passive and an active protocol to achieve r=0.01 infidelityto the target state. Similarly to the Fig. 2 for AKLT targetstate, note the clear advantage for each recorded run of theactive protocol compared to the mean duration ' 4.6 · 103

of the passive protocol. Each histogram was obtained from104 numerical simulations, and truncated at 15000 cycles forbetter presentation.

(largely equivalent to “blind” steering) state preparation.While our approach has sweeping applicability, here wehave chosen to focus on active measurement-driven steer-ing as applied to the most challenging case of many-bodyquantum systems with entangled target states.

To satisfy physical (locality) constraints on system-detector couplings, we have proposed a scheme, basedon parent Hamiltonian construction, for identifying feasi-ble couplings. Employing such couplings, we have devel-

16

oped and analyzed Hilbert-space-orientation techniquesfor measurement-driven steering. A central ingredienthere has been to develop feedback policies based on detec-tor readouts. One such Hilbert-space path-finding tech-nique is based on a cost function, evaluating the runningfidelity to the target state. We have shown a substantial(up to 9.5-fold) speedup of steering, employing this ap-proach for preparation of the ground state of the AKLTmodel. A second protocol comprises mapping out theavailable measurement actions onto a Quantum StateMachine (QSM), using a coarse-grained version of thecorresponding graphs in Hilbert space. We have given anexample of an entangled state preparation which showsacceleration by a factor of 12.5 compared to passive steer-ing.

While we have limited ourselves here to a few exam-ples, our schemes are of general applicability. They openthe door to the design of efficient and high-quality stateengineering, adiabatic state manipulation, and, possi-bly, quantum information processing. Moreover, steer-ing protocols are subject to errors, both “static” (choiceof steering parameters) and “dynamics” (noise). Activedecision-making steering may be designed to reduce theeffect of such errors. One may envision a host of direc-tions to generalize and develop these ideas. For example,the greedy minimization of our cost function may be fur-ther improved by finding other metrics of local “steepestdecent.” Further, one may systematically investigate lesslocal (less greedy) optimization of the cost function, e.g.,looking n steps ahead. Another potential advantage ofour protocols relies on the following observation: in thecontext of passive steering, one imposes constraints con-cerning locality (e.g., how many spins can be coupled toa local detector), and certain types of coupling terms.Given such constraints, not all target states are reach-able. The introduction of active steering may overcomethis handicap of target-state accessibility.

One may combine the dynamics incorporated here withthe inherent unitary evolution of the system at hand (dueto a system-only Hamiltonian). Consider the context

of passive (blind) measurement-induced steering, which,in the continuum time limit, leads to Lindbladian dy-namics. Then, the addition of Hamiltonian dynamicsenriches the variability of steering, allowing, for exam-ple, to obtain mixed states by design [37]. It is intrigu-ing to investigate how the addition of Hamiltonian dy-namics extends or improves active steering, thus marry-ing the frameworks of closed-loop quantum control forHamiltonian-based state preparation and active-decisionmeasurement-based steering.

Further extensions of our approach include applica-tions of QSM protocols to larger and more complex sys-tems, going beyond a three-qubit setup. Optimizing suchprotocols may involve automatization of the creation andanalysis of QSMs, e.g., for finding an optimal basis au-tomatically, in similarity with quantum annealing, butnow at the level of measurement operators. Finally, onemay envision using machine learning to find more opti-mized navigation protocols (see [38, 39] for related workin the context of Hamiltonian feedback and open-loopcontrol). Given the delayed-reward setting at hand, a re-inforcement learning strategy such as Q-learning [40] orSARSA [41] might be the most appropriate choice.

ACKNOWLEDGMENTS

We thank J. Chalker, R. Egger, C. Koch, P. Ku-mar, E. Medina Guerra, S. Morales, G. Morigi, S. Roy,K. Snizhko, S. Polla, X. Bonet-Monroig, and A. Zazunovfor discussions. The work was supported by the DeutscheForschungsgemeinschaft (DFG): Project No. 277101999– TRR 183 (Project C01) and Grants No. EG 96/13-1 and No. GO 1405/6-1, as well as by the Israel Sci-ence Foundation and the Helmholtz International FellowAward, the National Science Foundation through awardDMR-2037654, the US-Israel Binational Science Founda-tion (BSF), and the Netherlands Organization for Scien-tific Research (NWO/OCW).

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