Qiang, X., Zhou, X., Wang, J., Wilkes, C., Loke, T., O'Gara, S., Kling,L., Marshall, G., Santagati, R., Ralph, TC., Wang, J., O'Brien, J.,Thompson, M., & Matthews, J. (2018). Large-scale silicon quantumphotonics implementing arbitrary two-qubit processing. NaturePhotonics, 12(9), 534-539. https://doi.org/10.1038/s41566-018-0236-y

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Large-scale silicon quantum photonics implementing arbitrary two-qubit processing

Xiaogang Qiang,1, 3, 4 Xiaoqi Zhou,2, 1, ∗ Jianwei Wang,1 Callum M. Wilkes,1 Thomas Loke,5

Sean O’Gara,1 Laurent Kling,1 Graham D. Marshall,1 Raffaele Santagati,1 Timothy C. Ralph,6

Jingbo B. Wang,5 Jeremy L. O’Brien,1 Mark G. Thompson,1 and Jonathan C. F. Matthews1, †

1Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory andDepartment of Electrical & Electronic Engineering, University of Bristol, BS8 1FD,UK.

2State Key Laboratory of Optoelectronic Materials and Technologies and School of Physics,Sun Yat-sen University, Guangzhou 510275, China

3State Key Laboratory of High Performance Computing and School of Computer Science, NUDT, Changsha 410073, China4National Innovation Institute of Defense Technology, AMS, Beijing 100071, China

5School of Physics, The University of Western Australia, Crawley WA 6009, Australia6Centre for Quantum Computation and Communication Technology, Department of Mathematical and Physics,

The University of Queensland, St. Lucia, Queensland 4072, Australia(Dated: April 15, 2018)

Integrated optics is an engineering solution proposed for exquisite control of photonic quantum in-formation. Here we use silicon photonics and the linear combination of quantum operators scheme torealise a fully programmable two-qubit quantum processor, which enables universal two-qubit quan-tum information processing in optics for the first time. The device is fabricated with readily avail-able CMOS based processing and comprises four nonlinear photon-sources, four filters, eighty-twobeam splitters and fifty-eight individually addressable phase shifters. To demonstrate performance,we programmed the device to implement ninety-eight various two-qubit unitary operations (withaverage quantum process fidelity of 93.2±4.5%), a two-qubit quantum approximate optimizationalgorithm and efficient simulation of Szegedy directed quantum walks. This fosters further use ofthe linear combination architecture with silicon photonics for future photonic quantum processors.

The range and quality of control that a device has overquantum physics determines the extent of quantum in-formation processing (QIP) tasks that it can perform.One device capable of performing any given QIP taskis an ultimate goal1 and silicon quantum photonics2 hasattractive traits to achieve this: photonic qubits are ro-bust to environmental noise3, single qubit operations canbe performed with high precision4, a high density of re-configurable components have been used to manipulatecoherent light5,6 and established fabrication processes areCMOS compatible. However, quantum control needs toinclude entangling operations to be relevant to QIP —this is recognised as one of the most challenging tasksfor photonics because of the extra resources required foreach entangling step3,7. Here, we demonstrate a pro-grammable silicon photonics chip that generates two pho-tonic qubits, on which it then performs arbitrary two-qubit untiary operations, including arbitrary entanglingoperations. This is achieved by using silicon photonics toreach the complexity required to implement an iterationof the linear combination of unitaries architecture8,9 thatwe have adapted to realise universal two-qubit process-ing. The device’s performance shows that the design andfabrication techniques used in its implementation workwell with the linear combination architecture and can beused to realise larger and more powerful photonic quan-tum processors.

Miniaturisation of quantum-photonic experiments intochip-scale waveguide circuits started10 from the need torealise many-mode devices with inherent sub-wavelengthstability for generalised quantum-interference experi-ments, such as multi-photon quantum walks11 and bo-

son sampling12–14. Universal six-mode linear opticsimplemented with a silica waveguide chip (coupled tofree-space photon sources and fibre-coupled detectors)demonstrated the principle that single photonic devicescan be configured to perform any given linear opticstask15. Silicon waveguides promise even greater capabil-ity for large-scale photonic processing, because of theirthird order nonlinearity that enables photon pair gen-eration within integrated structures16, their capacity forintegration with single photon detectors17 and their com-ponent density can be more than three orders of magni-tude higher than silica2.

Programmable quantum processors have been reportedwith up to five trapped-ion qubits18, eleven NMRqubits19 and tens of superconducting qubits20. However,for photons, up to two sequential two-qubit entanglingoperations implemented with free-space optics21,22 andsilicon quantum photonics23,24 is the state of the art inqubit control. But the degree of control and utility ofthese photonic demonstrators is limited intrinsically be-cause arbitrary two-qubit processing requires the equiv-alent of three consecutive entangling gates in the circuitmodel of quantum computing, as demonstrated exper-imentally in 2010 with ion-trap quantum processing25.Effective QIP with three sequential entangling operationsis beyond the level of complexity that can be practicallyconstructed and maintained with free-space quantum op-tics or a hybrid of free-space nonlinear optics and inte-grated linear optics15.

Our scheme realizes arbitrary two-qubit unitaryoperation via a linear combination of four easy-to-implement unitaries — each being a tensor product

2

(1) (2) (3) (4) (5)

a

b

c

d

e

f

g

h

D

), QXG, 2017/07/12, try different style2017/08/30, modified: (A) and add a-f for paths in chip2017/09/28, remove the optical scheme figures2017/11/01, redraw LC circuits 2017/11/18, replace I, X, Y ,Z2017/11/23, update chip figure

， updated based on Jianwei’s comments，updated based on Xiaoqi’s comment

CoincidenceCountingPC

TunableLaser

DAC: 64 Channels; 0-20 mA; 10 kHz; 16 bit

EDFA DWDM

SNSPD

A ȁ 𝜑 ෑ𝑖=0

𝑘−1

𝑉𝑘−1−𝑖 ȁ 𝜑𝑉0 𝑉1 𝑉𝑘−1⋯

𝑉0ȁ 𝜑 𝑉1 𝑉𝑘−1

⋯

H

H

H

𝑖=0

𝑘−1

𝛼𝑖𝑉𝑖 ȁ 𝜑

ȁ 0

ȁ 0

ȁ 0

⋯ ULC𝑛

B

C H

Hȁ 0

ȁ 0ULC

ȁ 𝜑12I

I 𝜎𝑥

Q1

Q2

P1

P2

I

I

𝜎𝑥 𝜎𝑦

𝜎𝑦

𝜎𝑧

𝜎𝑧

𝜎𝑥 𝜎𝑦

𝜎𝑦

𝜎𝑧

𝜎𝑧𝜎𝑥

DEVICE

PC

Classical Processing

FIG. 1: Quantum information processing circuits and a schematic of the experimental setup. (A) The conventionalquantum circuit model of QIP, that is a multiplication of quantum logic gates in series. (B) Probabilistic linear-combination

of quantum gates. The operation∑k−1i=0 αiVi is implemented when all n control qubits are measured to be 0. ULC is a unitary

operation with first row in its matrix representation given as {α0, α1, · · · , αn−1} where∑k−1i=0 |αi|

2 = 1, k = 2n and thesuccess probability is 1/k. Other rows are chosen accordingly to make ULC unitary. (C) Deterministic linear-combinationcircuit for universal two-qubit unitary operation. For a U ∈ SU(4) being decomposed as Equation (1), ULC is defined as[α0, α1, α2, α3;α1, α0,−α3,−α2;α2,−α3, α0,−α1;α3,−α2,−α1, α0]. |ϕ12〉 is an arbitrary two-qubit state. The required twoauxiliary control qubits can also be replaced by a ququart (four-level system) and then ULC is a single-ququart operation.(D) Schematic of our device and external setup. A tunable continuous wave laser is amplified with an optical fibre amplifier(EDFA), spectrally filtered by a dense wavelength-division multiplexing (DWDM) module and launched into the device througha V-groove fibre array (VGA). Photons emerging from the device are collected by the same VGA and two DWDMs are used toseparate the signal (red) and idler (blue) photons. Photons are detected by two fibre-coupled superconducting nanowire single-photon detectors (SNSPD). The polarisations of input/output photons are optimised by in-line polarization controllers (PC).Coincidence counting logic records the two-photon coincidence events. Phase shifters on the device are configured througha digital-to-analog converter (DAC), controlled from a computer. The device includes five functional parts: (1) generatingququart-entanglement; (2) preparing initial single-qubit states; (3) implementing single-qubit operations; (4) realizing linear-combination; (5) changing the measurement basis. Part (1), (3) and (4), as a whole, are used to implement a given SU(4)operation, where part (1) encodes the linear-combination coefficients, part (3) implements linear terms Ai and Bi and part (4)realizes the linear combination of terms Ai ⊗ Bi together with post-selection. Part (2) prepares arbitrary separable two-qubitstates |ϕini〉 (= |ϕ1〉 ⊗ |ϕ2〉) as the input, which is independent of the implemented gate. Part (5) rotates the output state sothat it can be measured at desired basis.

of two single-qubit unitaries. The presented chip con-structs and exploits high-dimensional entanglement inorder to implement the equivalent capability of threesequential entangling gates in the circuit model whilstusing only two photons. It performs universal two-qubitprocessing with high fidelity whilst all thermal phaseshifters in the device are simultaneously active and itpossesses inherent phase stability of the optical pathsand waveguide interferometric structures. The chip isalso repeatable under continuous operation and it can

be reprogrammed at kilohertz rate. We demonstrate thechip’s performance by performing process tomographyon 98 implemented two-qubit quantum logic gates,by realising the quantum approximate optimizationalgorithm (QAOA)26,27 applied to three example con-straint satisfaction problems, and by simulating Szegedyquantum walks (SQW)28,29 over an example two-nodeweighted graph. All together, the results presentedrequired 98480 experiment configurations.

3

1. Linear combination of unitaries on a chip forQIP. The conventional quantum circuit model for QIP isa sequence of quantum gates (Fig. 1(A)). The linear com-bination of unitary operations is an alternative approach(Fig. 1(B)) that is central to various QIP tasks8,30–34.A universal two-qubit unitary U ∈ SU(4) can be imple-mented by the four-operator linear combination35

U =∑3i=0 αi (P1σiQ1)⊗ (P2σiQ2), (1)

where P and Q are single-qubit gates, σi are identity andPauli gates (I, σx, σy, σz) and αi are complex coefficients

satisfying∑3i=0 |αi|2 = 1. This linear combination can

immediately be implemented through two-qubit versionof the n-qubit circuit shown in Fig. 1(B), with an intrin-sic success probability of 1/4. However, we also note thata deterministic implementation of the linear-combinationof U can in principle be achieved with extra classical con-trolled gates35, as shown in Fig. 1(C). In the presentedchip, the linear decomposition of U is implemented prob-abilistically by expanding the dimension of qubits intoqudits and using pre-entanglement between qudit sys-tems that can be generated from parametric photon pairgeneration8,23. This Hilbert-space-expansion approachimplements arbitrary two-qubit unitaries using resourcesof only a two-photon entangled-ququart state and multi-mode interferometry35, that is inherently stable on ourchip.

Fig. 1(D) illustrates the schematic of our silicon pho-tonic chip operated with external electrical control,laser input and fibre coupled superconducting detec-tors. The 7.1 mm × 1.9 mm chip consists of fourspiral-waveguide spontaneous four-wave mixing (SFWM)photon-pair sources36, four laser pump rejection filters,eighty-two multi-mode interferometer (MMI) beam split-ters and fifty-eight simultaneously running thermo-opticphase shifters36. Within the device, the four SFWMsources are used to create possible (signal-idler) photonpairs when pumped with a laser that is launched intothe chip and split across the four sources according tocomplex coefficients αi. The spatially bunched photonpairs are coherently generated in either one of the foursources. Post-selecting when signal and idler photonsexit at the top two output modes (qubit 1) and the bot-tom two (qubit 2) respectively, yields a path-entangledququart state |Φ〉 as

α0|1〉a|1〉e+α1|1〉b |1〉f+α2|1〉c |1〉g+α3|1〉d |1〉h (2)

at the end of stage (1) marked on the device shown inFig. 1(D), with intrinsic success probability of 1/4. |1〉jrepresents the Fock state in spatial modes labeled by j =a, b, c, d, e, f, g, h.

Spatial modes a-h are each extended into two modesto form path-encoded single-qubit states |ϕ1〉 or |ϕ2〉with arbitrary amplitude and phase controlled withMach-Zehnder Interferometers (MZI) and an extra phaseshifter. Single-qubit operations Ai (=P1σiQ1) and Bi(=P2σiQ2) are applied using MZIs and phase shifters to

|ϕ1〉 and |ϕ2〉 respectively, evolving |Φ〉 into∑3i=0 αiAi |ϕ1〉ui Bi |ϕ2〉vi , (3)

where ui ∈ {a, b, c, d} and vi ∈ {e, f, g, h}. Next, thequbits a, b, c, d are combined into one final-stage qubit,and the qubits e, f, g, h into the remaining final-stagequbit, as shown in stage (4) of Fig. 1(D) with intrinsicsuccess probability 1/16. This removes path informationof the signal (idler) photon and thus we obtain the finalevolved two-photon state as(∑3

i=0 αiAi ⊗Bi)|ϕini〉 (4)

where |ϕini〉 (= |ϕ1〉 ⊗ |ϕ2〉) is an arbitrary separabletwo-qubit state. Once photons are generated, the overallintrinsic success probability of our design is 1/64, whichis higher than the two main schemes considered foruniversal linear optical quantum computation35: theKnill-Laflamme-Milburn (KLM) scheme7 and linearoptical measurement-based quantum computation(MBQC)37. The success probability of this opticalimplementation could be further increased to 1/4 if wewere to separate signal and idler photons with certaintyand use also an advanced linear-combination circuitthat utilizes the unused optical ports in our current chipdesign35.

2. Realising individual quantum gates. The linear-combination scheme can simplify implementation of fam-ilies of two-qubit gates. For example, an arbitrary two-qubit controlled-unitary gate CU can be implemented asthe linear combination of two terms:

CU = 1√2

(1 00 i

)⊗ I−iU√

2+ 1√

2

(1 00 −i

)⊗ I+iU√

2, (5)

and SWAP gate can be implemented by a linear combi-nation of only identity and Pauli gates:

SWAP =1

2(I ⊗ I + σx ⊗ σx + σy ⊗ σy + σz ⊗ σz) . (6)

To show the reconfigurability and performance of ourchip, we implemented 98 different two-qubit quantumlogic gates, for which we performed on-chip full quan-tum process tomography and reconstructed the processmatrix using the maximum likelihood estimation tech-nique for each gate35. A histogram of measured processfidelities for these 98 gates is shown in Fig. 2(A), witha mean statistical fidelity of 93.15±4.53%. The imple-mented gates include many common instances—as shownin Fig. 2(B, C)—achieving high fidelities, such as CNOTwith 98.85±0.06% and SWAP with 95.33±0.24%. Ourdevice also allows implementation of non-unitary quan-tum operations. The entanglement filter (EF)8,38 andthe entanglement splitter (ES)8 can be implemented by

EF = (I ⊗ I + σz ⊗ σz)/√

2 (7)

ES = (I ⊗ I − σz ⊗ σz)/√

2 (8)

4

The results are shown in Fig. 2(D) and (E) in the formof logical basis truth tables, with the classical fidelitiesof 95.31±0.45% and 97.69±0.31% respectively. A furtherdiscussion about the experimental fidelities is presentedin the Supplementary Information.

CSP1 CSP2 CSP3

CSP1:

CSP2:

CSP3:

Implementing QAOA

Simulating SQW

Realizing arbitrary 2-qubit gatesA B C

E

D

F

G

H

I

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K L

M N

Gate Fidelity

CNOT 98.85±0.06%

CZ 96.90±0.17%

CH 97.57±0.07%

SWAP 95.33±0.24%

iSWAP 94.45±0.27%

SWAP 92.41±0.33%____

A B

C

D E

Coincide

nce

Coun

t(×102/s) 20

10

0|00i

|01i|10i

|11i|00i |01i |10i |11i

|00i|01i

|10i|11i

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20

10

0

OutputInput

Coincide

nce

Coun

t(×102/s)

OutputInput

Prob

ability

OutputInput

Fidelity

Freq

uency

FIG. 2: Experimental realisation of arbitrary 2-qubitgates. (A) A histogram of measured process fidelities for 98two-qubit quantum gates (F = 93.15±4.53%). (B) Measuredprocess fidelities for example two-qubit gates: C-NOT, C-Z,C-H, SWAP, iSWAP,

√SWAP. The errors are estimated by

adding random noise to the raw data and performing manyreconstructions. (C) The real part of experimentally deter-mined process matrices of SWAP, with ideal theoretical valuesoverlaid. (D, E) Logical basis truth tables for entanglementfilter (D) and entanglement splitter (E).

3. Implementing a two-qubit Quantum Approxi-mate Optimization Algorithm for Constraint Sat-isfaction Problems. The quantum approximate op-timization algorithm (QAOA) was proposed for findingapproximate solutions to combinatorial search problemssuch as constraint satisfaction problems (CSPs)26,27. It isa promising candidate to run on primitive quantum com-puters because of its possible use for optimization andits conjectured potential as a route to establishing quan-tum supremacy39. A general CSP is specified by n bitsand a collection of m constraints—each of which involves

CSP1 CSP2 CSP3

CSP1:

CSP2:

CSP3:

Implementing QAOA

Simulating SQW

Realizing arbitrary 2-qubit gatesA B C

E

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CSP1 CSP2 CSP3

CSP1:

CSP2:

CSP3:

Implementing QAOA

Simulating SQW

Realizing arbitrary 2-qubit gatesA B C

E

D

F

G

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M N

C

C

B

B

H

H

|0i|0i Ce

|0i|0i Ce

B

B

H

H

C

C

B

B

H

H

|0i|0i C

0e

CSP1 CSP2 CSP3A

D

B

C

FIG. 3: Experimental realisation of a two-qubit quan-tum approximate optimisation algorithm. Panels ar-ranged into three columns, corresponding to three exampleCSPs, labeled 1-3. (A) Quantum circuits of QAOA for eachCSP. (B) Theoretical and (C) experimentally determined val-ues of 〈γ, β|C|γ, β〉 over the grid of [γ, β] ∈ [0, 2π]× [0, π] forCSP1, CSP2 and CSP3, with step size δγ = 2π

20, δβ = π

30, for

finding the optimized |γ, β〉 states. (D) Experimental mea-surement results of the optimized |γ, β〉 states, outputtingthe searched target string z for each CSP 1-3. The s.d. ofeach individual probability is calculated by propagating errorassuming Poissonian statistics.

a small subset of the bits. For a CSP, QAOA outputsa binary string z which (approximately) maximizes thenumber of satisfied constraints, i.e., C(z) =

∑ml=1 Cl(z)

where Cl(z) = 1 if z satisfies the l-th constraint, other-wise 0 — this is the goal of CSP.

The QAOA process can be summarised as follows.Suppose two operators C and B are defined as

C |z〉 := C(z) |z〉 , B :=∑ni=1 σ

(i)x (9)

where σ(i)x represents σx acting on the i-th qubit, and a

quantum state |~γ, ~β〉 is defined as

|~γ, ~β〉 = e−iβpBe−iγpC · · · e−iβ1Be−iγ1CH⊗n |0〉⊗n (10)

where ~γ := (γ1, · · · , γp) ∈ [0, 2π]p and ~β :=(β1, · · · , βp) ∈ [0, π]p. QAOA seeks the target string z

by searching the ~γ and ~β that maximize 〈~γ, ~β|C|~γ, ~β〉and then the corresponding state |~γ, ~β〉 in the computa-

tional basis encodes the solution. For a given ~γ and ~β,

〈~γ, ~β|C|~γ, ~β〉 can be evaluated through a quantum com-puter, which can further be used as a subroutine in an en-veloping classical algorithm—for example, run the quan-

tum computer with angles (~γ, ~β) from a fine grid on the

5

CSP1 CSP2 CSP3

CSP1:

CSP2:

CSP3:

Implementing QAOA

Simulating SQW

Realizing arbitrary 2-qubit gatesA B C

E

D

F

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A

B

E

C D

F

FIG. 4: Experimental quantum simulation of Szegedy directed quantum walks. (A) A weighted two-node graph.Edge weights are decided by α, β ∈ [0, 1]. (B) Quantum circuit for a single-step SQW on the two-node graph. R(θ) is definedas R(θ) = [cos θ,− sin θ; sin θ, cos θ] with θ ∈ {θ1,−θ1, (θ1 − θ2), (θ2 − θ1)} where θ1 = arccos(

√1− α) and θ2 = arccos(

√β).

(C-F) Theoretical (red points) and experimental (gray bars) probability distributions (of the walker being at node 1) of SQWswith the initial state |00〉: (C) α = β = 0.25, Favg = 98.46± 0.04%; (D) α = β = 0.5, Favg = 98.48± 0.04%; (E) α = β = 0.43,Favg = 98.02± 0.04%; (F) α = 0.1, β = 0.9, Favg = 98.35± 0.15%. The s.d. of each individual probability is also plotted, whichis calculated by propagating error assuming Poissonian statistics.

set [0, 2π]p×[0, π]p—to find the best ~γ and ~β for maximiz-

ing 〈~γ, ~β|C|~γ, ~β〉39. With p getting increased, the qualityof the approximation of QAOA improves26.

In our experiments, we restricted to the p = 1 caseof QAOA, and applied QAOA to three 2-bit CSPs. Thecorresponding quantum circuits are shown in Fig. 3(A).The first CSP (denoted as CSP1) is the 2-bit Max2Xorproblem which has just one constraint as C(z) = 1

2 +12z1z2 where z1, z2 ∈ {±1}. The other two CSPs havethree constraints:

CSP2 : (11)

C1(z) = 12 + 1

2z1;C2(z) = 12 + 1

2z2;C3(z) = 12 + 1

2z1z2

CSP3 : (12)

C1(z) = 12 + 1

2z1;C2(z) = 12 + 1

2z2;C3(z) = 12 − 1

2z1z2

For the p = 1 QAOA, there are only two angles, γ andβ, to be found for optimizing 〈γ, β|C|γ, β〉. We searchγ and β along a fine grid on the compact set [0, 2π] ×[0, π] and show each obtained value of 〈γ, β|C|γ, β〉 asin Fig. 3(B,C) where the target angles are marked asthe reddest block. By measuring the corresponding|γ, β〉 state in the computational basis for CSP1, we ob-tain “00” or “11” with highest probability, correspond-ing to the target string of CSP1: {z1, z2} = {1, 1} or{−1,−1}. Similarly, the obtained results for CSP2 is{z1, z2} = {1, 1} and for CSP3 are {z1, z2} = {1, 1},{1,−1} or {−1, 1}—either of which is a solution of CSP3.The experimental results are shown in Fig. 3(D), with

the classical fidelities between experiment and theoryof 99.88±0.10%, 96.98±0.56% and 99.48±0.27% respec-tively.

4. Simulating Szegedy Quantum Walks. Quan-tum walks model a quantum particle’s random movementin a discretized space according to a given set of rulesknown as a graph. They are of interest for developingquantum computing (e.g. Ref. 40) and quantum algo-rithms (e.g. Ref. 41) and as an observable quantum phe-nomena11. The Szegedy quantum walk (SQW)28,29 is aparticular class that allows unitary evolution on directedand weighted graphs—which the standard discrete-timeand continuous-time quantum walk formalisms do notpermit—and has been proposed for application to quan-tum speedup for ranking the relative importance of nodesin connected database42–44. The realization of SQW-based algorithms on a quantum computer requires anefficient quantum circuit implementation for the walk it-self45,46. Here we have implemented SQW experimen-tally on an example two-node graph with four weighteddirected edges.

A general weighted graph G with N nodes can be de-scribed by its transition matrix P where an element Pi,jis given by the weight of a directed edge from node i to

j, satisfying∑N−1i=0 Pi,j = 1. A SQW on G is defined as

a discrete-time unitary time evolution on a Hilbert spaceH = H1⊗H2 where H1 and H2 are both N -dimensionalHilbert spaces, and thus its quantum circuit implementa-tion requires 2 logN many qubits. The single-step oper-

6

ator of an SQW is given by Usz = S(2Π− I). Here S is a

SWAP operator defined as S =∑N−1i=0

∑N−1j=0 |i, j〉 〈j, i|,

and Π is a projection operator as Π =∑N−1i=0 |φi〉 〈φi|

with |φi〉 = |i〉⊗∑N−1j=0

√Pj+1,i+1 |j〉 for i ∈ {0, · · · , N−

1}. For the example two-node graph that we label E andsketched in Fig. 4(A), a quantum circuit implementationfor single-step SQW operator can be constructed by us-ing the scheme proposed in ref46, as shown in Fig. 4(B).Repeating this circuit generates an efficient quantum cir-cuit implementation of multiple-step SQWs, which caneasily simulate the dynamics of SQWs on the examplegraph and its variants.

The periodicity of SQWs is determined by theeigenvalues of the single-step operator Usz, andit has been studied on several families of finitegraphs47. Usz of the graph E has four eigenvalues:{−1, 1, 1 − s −

√s2 − 2s, 1 − s +

√s2 − 2s} where

s = α + β and α, β ∈ [0, 1]. Usz is periodic if and onlyif there exists an integer n such that λni = 1 for allfour eigenvalues λi of Usz. The period is then givenby the lowest common multiple of the periods of theeigenvalues. E has a symmetric transition matrix whenα = β. For SQWs on a symmetric graph E , periodicwalks exist in the cases α = β = 1

4 ,12 ,

34 , 1—with periods

of 6, 4, 6 and 2 steps respectively—of which the first twoare experimentally verified as shown in Fig. 4(C) and(D). SQWs on a general instance of E do not exhibitperfect periodicity, as shown by Fig. 4(E) that showsthe behaviour of SQWs on E with α = β = 0.43. Ehas an asymmetric transition matrix when α 6= β, andperfect periodicities of SQWs can exist in particularcases, such as α + β = 1, which has a period of 4 steps.An example of this kind, α = 0.1, β = 0.9, is shownin Fig. 4(F). In our device, we can also perform statetomography on a given time-evolved state of SQWs —we have performed quantum state tomography for morethan 500 time-evolved states, observing an average statefidelity of 93.95±2.52% with theoretical prediction.

5. Discussion. The computational capacity of thelinear-combination protocol can be expanded by increas-ing the dimensionality of each linear-combination term.

The high-dimensional quantum state can be easily pre-pared, manipulated and measured with Reck et al.- stylelinear optical network15. For example, a four-qubit To-folli gate can be effectively implemented through a four-qubit version of the linear-combination protocol thatutilizes a two-photon six-dimensional entanglement andfour-dimensional Reck et al.- style circuits, as illustratedin the Supplementary Information. However, the linear-combination protocol has its limitation on scaling up foruniversal quantum computation, as its success probabil-ity is inversely proportional to the number of terms. Al-though the protocol cannot be targeted for the ultimatequantum computer, it possesses great potential in thenear- and mid-term for situations where the photoniccomponents are easier to create than the photons them-selves and it is no less demanding of individual compo-nent performance than other linear optics approaches toQIP. Our range of demonstrations with a single devicehas shown that the linear-combination scheme is valu-able in permitting QIP demonstrations with the currentstate of the art in photonics and that silicon photonicsis capable of fulfilling its requirements. The device re-ported comprises nonlinear photon sources, optical filter-ing and reconfigurable linear optics and it was fabricatedwith a standard CMOS based silicon photonics processesonto a single photonic chip. It generates photons, en-codes quantum information on them, manipulates themand performs tomographic measurement, all with highfidelity quantum control for thousands of configurations.From our experience, our demonstrations of the QAOAand SQWs are beyond the practicality and performanceachievable with free-space bulk optical experiments andglass-based integrated photonics. Together with devel-oping multi-photon sources48 and integration with onchip detection49, future iterations of silicon photonicsopens the way to more sophisticated photonic experi-ments that are impossible to achieve otherwise, includingthe eventual full-scale universal quantum technologies us-ing light50.

Data access statement: The data that support thefindings of this study are available from the correspond-ing author upon reasonable request.

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