University of LjubljanaFaculty for Mathematics and Physics

Department for Physics

SEMINAR - 4.letnik

ADIABATIC QUANTUM COMPUTATION

Avtor: Enej Ilievskiemail: enej.ilievski@student.fmf.uni-lj.si

Mentor: dr. Marko Žnidarič

May 17, 2010

Abstract

In this seminar we present adiabatic quantum algorithm – a quantum computational methodfor solving hard computational problems which relies on the adiabatic theorem. Such approach isparticularly interesting because it may offer possibilities to reduce the effects of quantum decoherence.

We are mainly focusing on the conceptual background.

1 INTRODUCTIONIn adiabatic quantum computation (AQC) one encodes a computational problem in a suitable physicalsystem in such way that the structure of the ground (lowest energy) state reveals the answer to theproblem. In order to find the ground state one starts with engineering some simple Hamiltonian inits ground state and gradually deforms it into complex Hamiltonian whose ground state encodes thesolution to the problem. If this deformation is sufficiently gradual, then the transformation of the stateis adiabatic, and the system remains in its ground state throughout the evolution. AQC is known to bean universal model of quantum computation.

We begin this seminar by briefly describing some basic concepts in quantum computation, espe-cially related to the standard (logic gate) quantum computer, following by a short introduction to thepossibilities of implementing quantum computers and comparing their power with respect to classicalcomputers.

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Before focusing on the adiabatic quantum computation and quantum annealing principles, we presentsome alternative approaches in quantum computation.

At the end we present some recent theoretical and practical developements together with practicalexperimental implementation achievements concerning AQC.

Although we give a short introduction to quantum computing some basic knowledge of quantummechanics is required for proper understanding of this seminar.

2 QUANTUM COMPUTERQuantum computer is a device for computation that uses quantum mechanics phenomena (such assuperposition of states and quantum entanglement) to perform operations on data. A theoretical modelis an abstract machine, called quantum Turing machine (or universal quantum computer), which has thesame relation to quantum computation that normal Turing machines have to classical computation [1].1

2.1 BITS VS. QUBITSA memory of classical computer consists of well-known bits, where each bit represents either a one or azero. Quantum computer, on the other hand, performs operations on qubits (Fig. 1). Qubit is formallya two-level quantum mechanical system where one eigenstate refers to value 0 (or |0〉 in Dirac notation)and other to value 1. Quantum systems can be found in superposition of states, thus a qubit is given by

|ψ〉 = α|0〉+ β|1〉.

Figure 1: Qubit, a fundamental entity in quantum informatics, can be represented as a point on Blochsphere; |ψ〉 = cos (θ/2)|0〉 + eiφ sin (θ/2)|1〉. A state of qubit is a vector in two-dimensional complexvector (Hilbert) space. Special states |0〉 and |1〉 are known as computational basis states and form anorthogonal basis of this vector space.

The crucial fact is, that we cannot examine a qubit to determine its quantum state, i.e. coefficientsα and β. In fact, when we measure a qubit we get either the result 0, with probability |α|2, or result1, with probability |β|2. 2 After the measurement a qubit is found in a state that corresponds to themeasurement result, e.g. if we get 0, |ψ〉 “collapses” to state |0〉.

Moreover, a pair of qubits can be found in a superposition of 4 states (|00〉, |01〉, |10〉, |11〉), threequbits in 8 states, or in general, n qubits in 2n states 3 (n classical bits represent only one of possible2n states). The computational basis vectors of n-qubit system are of the form |x1x2 . . . xn〉, xi ∈ {0, 1},

1It means that any quantum algorithm can be represented as particular quantum Turing machine.2Of course, |α|2 + |β|2 = 1 holds.3Unfortunately there is no simple generalization of Bloch sphere for multiple qubits.

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so a quantum state of such a system is specified with 2n amplitudes. For n = 300 this number is largerthan the estimated number of atoms in the Universe! Trying to store all this information on classicalcomputer would be impossible. However, it seems Nature is capable of manipulating such enormousquantities of data during evolution of quantum systems and this huge computational power is reallysomething we would like to take advantage of.

Many physical systems can be used to represent qubits, for example two different polarizations of aphoton, alignment of a nuclear spin in uniform magnetic field, two states of an electron orbiting a singleatom etc.

2.2 QUANTUM COMPUTER USING QUANTUM LOGIC GATESUndoubtly the most popular model for quantum computation is a quantum circuit model in whicha computation is a sequence of reversible transformations on n-qubit (quantum) register. 4 Thesetransformations are unitary (probability preserving) and are called quantum gates [2, 3].

An arbitrary quantum computation (reversible operation) on any number of qubits can be generatedby a finite set of quantum gates. Such set is said to be universal for quantum computation. An universalgates for classical computation are NAND and NOR gates. 5 In quantum computation, any multiplequbit logic gate can be composed from CNOT (2-qubit gate) and single qubit gates (described by 2× 2unitary matrices) such as Hadamard gate and phase gate. 6

As we intuitively expect, quantum computers can simulate classical computations (non-deterministicas well), but there would be little point in going to all the trouble with quantum effects if we would notbe able to solve some problems much more efficiently than classical computers.

Broadly speaking, there are three classes of quantum algorithms which provide an advantage overknown classical algorithms.

1. A class of algorithms based upon quantum version of the Fourier transform (Shor’s algorithm forfactoring and discrete logarithm).

2. Quantum search algorithms (Grover’s algorithm 7).

3. Quantum simulations (algorithms for simulatiing quantum systems 8).

2.3 QUANTUM DECOHERENCEOne of the greatest challenges in quantum computation is controlling or removing quantum decoherencewhich usually means isolating system from its environment. Quantum decoherence is a consequence ofinteraction of quantum systems with their environments resulting in their probabilistic behaviour. 9

This is a non-unitary effect (irreversible) and can be viewed as the loss of infomation from the systemto environment.

4Note that classical logic gates are irreversible (non-invertible)!5Universal classical computation can also be done combining only 3-bit Toffoli gates.6A single-gate universal quantum gates can also be formulated using 3-qubit Deutsch gate.7Given a search space of size N (with no prior knowledge about the structure of information in it) we want to find the

element satisfying a known property. Classicaly, this problem can be solved in O(N), but the quantum search algorithmcan do it using only O(

√N) operations.

8The main difficulty of simulating quantum systems on classical computers is exponentially growing size of the system(number of complex numbers needed to describe it). By contrast, a quantum computer can perform the simulation usingO(n) qubits, however the question how to efficiently extract desired information from the quantum system still remains.

9Note that the combined system (system + environment togehter) is always in a pure quantum state and desribed byunitary evolution. However, due to interacions of with external (unknown) degrees of freedom, system behaves as statisticalensemble of different states, rather than quantum superposition of them.

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2.4 PHYSICAL REALIZATION OF QUANTUM COMPUTERTo realize a quantum computer, we must not only give qubits some robust physical representation, butalso select a system in which they can be made to evolve as desired. Furthermore, we must be able toprepare qubits in some specified set of initial states, and to measure the final output.

A single nuclear spin can be a good choice for a qubit, because superpositions of being aligned with oragainst an external magnetic field can last a long time. Unfortunately it is difficult to build a quantumcomputer from nuclear spins because their coupling to the world is so small that it is hard to measurethe orientation of a single nuclei.

Physical support Name Information support |0〉 |1〉Single photon Polarization encoding Polarization of light Horizontal Vertical

Photon number Photon number Vacuum Single photon stateElectrons Electronic spin Spin Up Down

Electron number Charge No electron One electronNucleus Nuclear spin (NMR) Spin Up Down

Optical lattices Atomic spin Spin Up DownJosephson junction SC charge qubit Charge Uncharged SC island Charged SC island

(Q = 0) (Q = 2e, extra Cooper pair)SC flux qubit Current CW current CCW currentSC phase qubit Energy Ground state First exicited state

Singly chargedquantum dot pair Electron localization Charge Electron on left dot Electron on right dotQuantum dot Dot spin Spin Down Up

Table 1: Incomplete list of physical implementations of qubits (choices of basis are by convention)[4].

Despite huge number of entries in the table above, only three fundamentally different qubit repre-sentations exist: spin, charge and photon.

A key concept in understanding the merit of a particular realization is the notion of quantum noise(sometimes called decoherence). This is because the length of the longest possible quantum computationis given by the ratio of decoherence time 10 to operation time 11. These two times are determined by thestrength of coupling of the system to the external world.

Still, if the error rate (due to decoherence) is small enough, it is thought to be possible to use quantumerror correction, thereby allowing the total calculation time to be longer than decoherence time (the costis greatly increased number of required qubits).

The key idea is that if we wish to protect a system against the effects of noise we have to incorporatesome redundant information. Protecting bits against the effects of noise is somehow trivial in classicalworld, where for instance one needs to simply replace each bit with three copies of itself. Similarprocedure is not applicable to quantum systems due to no-cloning theorem, however there are twoadditional important issues:

1. Errors are continuous - different types of continuous errors may affect the state on a single qubit.Determining which error has occured in order to correct it would require infinite precision andresources.

2. Destructive measurements - observation in quantum mechanics destroys the quantum state underobservation, thus making it’s recovery impossible.

Fortunately these problems are not always fatal. Basic principles of error-correcting procedures arepresented in [5].

10The time for which the system remains quantum-mechanically coherent.11The time it takes to perform elementary unitary transformations (at least two qubits).

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2.5 THE POWER OF QUANTUM COMPUTATIONHow powerful quantum computers really are? Nobody yet knows the answer to this question, despitesome examples (such as factoring) suggest that quantum computers are more powerful than classicalcomputers. It is still possible that quantum computers are no more powerful than classical ones, in thesense that any problem which can be efficiently solved on a quantum computer can be also efficientlysolved on a classical computer.

Computational complexity theory is the subject of classifying the difficulty of various computationalproblems (both classical and quantum). The basic concept is a complexity class, which can be thoughtas a collection of computational problems that share some common feature(s) with respect to the com-putational resources needed to solve those problems.

The most important classes are P and NP. The former is the class of computational problems thatcan be solved efficiently (in polynomial time) on classical computer and the later is the class of problemswhich have solutions that can be quickly verified (again in polynomial time) on classical computer. It isclear that P is a subset of NP, since the ability to solve problem implies the ability to check potential forsolutions. Perhaps the most important problem of theoretical computer science is to determine whether

these two classes are different: P?

6=NP. 12

There is an important subclass of NP problems, called NP-complete problems (NPC). Any NPCproblem is at least as hard as all other problems in NP. It means that an algorithm to solve a specificNPC problem can be adapted 13 to solve any other problem in NP. If P 6=NP, then it follows that noNP-complete problem can be efficiently solved on classical computer.

It is not known wheter quantum computers can be used to quickly solve all the problems in NP(although they can be used to solve some of them, e.g. factoring, which is believed not to be in P). 14

Another important class in PSPACE. It consist of problems which can be solved using resourceswhich are few in spatial size, but not necessary in time. PSPACE is believed to be strictly larger thanP and NP (see Fig. 2), although this has never been proved.

Finally we mention BPP complexity class containing problems that can be solved using randomizedalgorithms in polynomial time, if a bounded probability of error is allowed.

What about quantum complexity classes? We can define BQP (acronym stands for bounded error,quantum, polynomial time) to be the class of all computational problems which can be solved efficientlyon a quantum computer, allowing a bounded probability of error. Quantum computers run only proba-bilistic algorithms 15 , so BQP on quantum computer is the counterpart of BPP on classical computers.Exactly where BQP fits with respect to P, NP and PSPACE is not known. What is known is thatquantum computers can solve all the problems in P efficiently, but there are no problems outside ofPSPACE which can be solved efficiently, therefore BQP probably lies somewhere between them. Animportant implication is that if it is proved that quantum computers are strictly more powerful thanclassical computers, then it will follow that P is not equal to PSPACE.

Although quantum computers may be faster than classical computers they can’t solve any problemsthe classical computers can’t (given enough time and memory). Since a probabilistic Turing machine cansimulate quantum computers, they could never solve an undecidable problem like the halting problem.

12Most scientists believe there are problems in NP that are not included in P.13More precisely, any NP problem is polynomial time reducible to NPC problem.14Note that factoring is not known to be NP-complete, otherwise we would already know how to efficiently solve all the

problems in NP on a quantum computer.15However we can significally reduce the probability of error by repeating an algorithm several times.

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Figure 2: The suspected relationship between classical and quantum complexity classes. Where quantumcomputers fit between P and PSPACE is not known, in part of because we do not even know whetherPSPACE is bigger than P!

2.6 OTHER QUANTUM COMPUTER TYPESIt is important to stress that quantum computer based on quantum logical gates (also known as standardquantum computer) mentioned above is not the only possible way of performing quantum computations!Several other approaches have been proposed so far:

• One-way quantum computerIn 2001, Robert Raussendorf and Hans J. Briegel presented a scheme of quantum computationthat consist entirely of one-qubit measurements on a particular class of entangled states calledcluster states 16. The measurements are used to imprint a quantum logic circuit on the state (seeFig. 3). As the computation proceeds, the entanglement in the resource cluster state is progressivelydestroyed. Cluster states are thus one-way quantum computers and the measurements form theprogram (they replace the unitary evolution) [6, 7].

Figure 3: After initially creating a multiparticle entangled cluster state, a sequence of adaptive single-particle measurements is carried out. In each step of the computation, the measurement basis of thenext qubit depends on the specific program and on the outcome of previous measurement results [6].

• Quantum cellular automataIt refers to any of several models of quantum computation, which have been devised in analogy

16In quantum computation, a cluster state is a type of highly entangled state of multiple qubits. Cluster states can becreated efficiently in any system with a quantum Ising-type interaction between two-state particles in a lattice configuration.

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to conventional models of cellular automata 17 introduced by von Neumann. It may also refer toquantum dot cellular automata, which is a proposed implementation of classical cellular automataexploiting quantum mechanical phenomena.

The computation is considered to come about by parallel operation of multiple cells. These areusually taken to be identical quantum systems, e.g. qubits. Cells togheter form a (usually regular)network. The evolution of the system has several symmetries. The most important are locality(next state of the cell depends only on current state and that of its neighbours) and homogeneity(the evolution is indepentendt of time and acts the same everywhere). The state space of the cells,as well as the operations performed on them, should be motivated by the principles of quantummechanics [8].

• Topological quantum computerIt is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons18, whose world lines 19 cross over one another to form braids in a 3D spacetime (one temporalplus two spatial dimensions). These braids act like the logic gates that make up the computer andare described in terms of braid group [10]. The main advantage of a quantum computer based onquantum braids is its stability. While the smallest perturbations can cause a quantum particle dodecohere (introducing errors in the computation) they do not change the topological properties ofthe braids [11].

In a key developement for topological quantum computers, in 2005 Vladimir J. Goldman et.al.were said to have created the first experimental evidence of using fractional quantum Hall effectto create actual anyons [12].

Topologial quantum computers are equivalent in computational power to other standard modelsof quantum computation.

To learn more about quantum topological computer we refer the reader to [13].

• Adiabatic quanum computerFinally we mention adiabatic quantum computation which was initiated in 2001 by Edward Farhiet.al. Authors suggested a novel quantum algorithm for solving classical optimization problemssuch as satisfiability 20 (SAT) based on adiabatic theorem.

In the remaining of this seminar we are focusing on the adiabatic quantum computer and presentingit in more detail.

3 ADIABATIC QUANTUM COMPUTATIONAdiabatic quantum computation relies on adiabatic theorem to do calculations. The goal is to find aHamiltonian whose ground state corresponds to the solution of the problem of interest. First, a systemwith a simple Hamiltonian is taken and initialized to its ground state. Finally, the simple Hamiltonianis adiabatically evolved to the desired Hamiltonian. By the adiabatic theorem, the system remains inthe ground state, so that the final state of the system describes the solution to the problem.

17The cellular automata is an abstract system consisting of uniform (finite or infinite) grid of cells. Each of these cells canonly be in one of finite number of states, which are determined by its adjacent cells (neighbourhood). The most popularexample is known as “The Game of Life”[9].

18Anyons are neither fermions nor bosons, but they share characteristics of fermions that they cannot occupy the samestate. In the real world anyons emerge from the excitations in an electron gas in a very strong magnetic field, and carryfractional units of magnetic flux in a particle-like manner (fractional quantum Hall effect).

19In physics, the world line of an object in the unique path of that object as it travels through 4D timespace.20Satisfiability is the problem of determining if variables of a given Boolean formula can be assigned in such a way as to

make the formula evaluate to TRUE.

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Adiabatic quantum computation offers possibility to avoid the problem of quantum decoherece. Sincethe system is in the ground state, interference with the outside world cannot make it move to a lowerstate. The only thing we need to ensure is to keep the temperature of the bath (energy of envoronment)lower than the energy gap between ground and first exited state of the system. If this condition isfulfilled, the system has a very low probability of going to a higher energy state. Thus the system canin principe stay coherent as long as needed. 21

3.1 ADIABATIC THEOREMAdiabatic theorem simply states that a physical system remains in its instantaneous eigenstate if a givenperturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest ofHamiltonian spectrum.

Note that the term adiabatic is traditionally used in thermodynamics to describe processes withoutthe exchange of heat between the system and environment. The quantum mechanical definition issomehow closer to the thermodynamical concept of quasistatic process 22, and has no direct relationwith heat exchange. A true analogy comes when entropy (TD system) and quantum number (QMsystem) are both considered to remain unchanged in adiabatic processes.

Suppose an energy of the system H(t0) is given at some initial time t0, with the correspondingeigenstate labelled as ψ(x, t0). Changing conditions in a continuous manner we end up with a finalHamiltonian H(t1) at some later time t1 in a final state ψ(x, t1). The adiabatic theorem states that themodification of the system critically depends on time difference τ = t1 − t0.

Diabatic process Rapidly changing conditions prevent the system from adapting its configurationduring the process, hence the probability density remains unchanged. Diabatic passage (infinitely rapid)is reached in the limit τ → 0: |ψ(x, t1)|2 = |ψ(x, t0)|2.

Adiabatic process For a trully adiabatic process we require τ → ∞; in this case ψ(x, t1) will be aneigenstate of the final Hamiltonian H(t1), with a modified configuration: |ψ(x, t1)|2 6= |ψ(x, t0)|2.

Example: Quantum harmonic oscillator Let us quickly explain the effects of the adiabatic theoremon the simple example. Consider a quantum harmonic oscillator as the spring constant k is increased(resulting in a narrowing of the potential well in the Hamiltonian).

If k is increased adiabatically (dkdt → 0) then the system at time t will be in an instantaneous eigenstateof the current Hamiltonian H(t), corresponding to the initial eigenstate of H(0). In particular case, thismeans the quantum number associated with quantum harmonic oscilator will remain unchanged (ifsystem is initially in its ground state n = 1, remains in the ground state as potential is compressed).

For a rapidly increased spring constant, the system undergoes a diabatic process (dkdt →∞) in whichthe system has no time to adapt its functional form to the changing conditions. While the final statemust look identical to the initial state (|ψ(t)|2 = |ψ(0)|2), there is no eigenstate of the new HamiltonianH(t) that resembles the initial state, thus the final state is composed of a linear superposition of differenteigenstates of H(t) that reproduce the form of the initial state.

3.2 QUANTUM ANNEALINGQuantum annealing (abbrev. QA) is a general method for finding the global minimum of a givenobjective (cost) function over a given set of candidate solutions (the search space). It is used mainly for

21In reality problems with decoherence cannot be completely avoided as we shall see later on.22A quasistatic process is a TD process that happens infinitely slowly. In practice, it must be carried out on a time-scale

that is much longer than the relaxation time of the corresponding system.

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the problems where the search space is discrete (combinatorial optimization problems) with many localminima.

In quantum annealing, a current state is replaced by randomly selected “neighbour” state if thelatter has lower energy (value of the objective function). The process is controlled by the tunnelingfield strength, the parameter that determines the extent of the neighborhood of states explored by themethod. The tunneling field starts high (so that the neighborhood extends over the whole search space)and is slowly reduced through the computation.

Method is actually derived from its classical analogue called simulated annealing, where “temperature”parameter plays similar role to QA’s tunneling field strength. However in simulated annealing theneighborhood stays the same throughout the search and the temperature determines the probability(given by Boltzmann distribution) of moving to a higher energy state, while in QA the tunneling fieldstrength determines the neighborhood radius. 23

It has been demonstrated experimentally as well as theoretically, that QA can out rate thermalannealing in certain cases, specially, where the potential energy landscape consist of very high but thinbarriers, surrounding shallow local minima - it is very unlikely for the thermal fluctuations to get thesystem out of such local minima, while quantum tunneling probabilities depends not only on the heightof barrier, but also on its width.

Figure 4: Schematic view on quantum annealing - while optimizing the cost function of computationallyhard problem one has to get out of a shallower minimum in order to reach a deeper minimum. Classicallyone has to jump over the energy of the cost barriers separating them, while quantum mechanically onecan tunnel thought the same. If the barrier is high enough, thermal jump becomes very difficult, however,if the barrier is narrow enough, quantum tunneling becomes quite easy [14].

In practice, the optimization problem is encoded in Hamiltonian HP . The algorithm starts byintroducing strong quantum fluctuations by adding a disordering Hamiltonian H ′ that does not commutewith H,

H = HP + ΓH ′,

where Γ changes from one to zero during the evolution, thus slowly removing the disordering part. Γindeed plays a role of so called tunneling field strength. If the process is slow enough, the system willsettle in a local minima close to exact solution (the slower the process the better the solution will beachieved). The performance of the computation is conditioned by the residual energy (the distance from

23In more elaborate simulated annealing variants (such as adaptive simulated annealing), the neighborhood radius isalso varied using temperature value.

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exact solution) versus evolution time. The computation time is the time required to generate a residualenergy below some acceptable threshold value.

The main difference between QA and AQC is that in the latter, the system is constrained to itsground state at all times, starting from the ground state of initial Hamiltonian and ending in the groundstate of HP . In other words, AQC is an exact algorithm, while QA is heuristic.

3.3 ADIABATIC QUANTUM EVOLUTION AS A COMPUTATIONALTOOL

Quantum system evolves in time according to Schröedinger equation

id

dt|ψ(t)〉 = H(t)|ψ(t)〉,

where |ψ(t)〉 is the time-dependent state vector and H(t) is the time-dependent Hamiltonian operator(here we set ~ = 1). A quantum algorithm can be viewed as a specification of a Hamiltonian H(t) andinitial state |ψ(0)〉. These are chosen so that the state at time T (|ψ(T )〉) encodes the answer to theproblem at hand.

In designing our quantum algorithm we rely on the above introduced adiabatic quantum theorem.Adiabatic evolution refers to the situation where H(t) is slowly varying. Suppose evolution starts attime t = 0 in |ψg(0)〉 which is the ground state of Hamiltonian H(0). The adiabatic theorem guaranteesthat evolving state vector |ψ(t)〉 remains close to instantaneous ground state |ψg(t)〉 if H(t) varies slowlyenough.

To specify our algorithm we have to provide H(t) for 0 ≤ t ≤ T , where T denotes the runningtime of the algorithm. We choose H(t) so that the ground state of H(0) is known in advance and easyto construct. For any particular instance of the problem, there is a Hamiltonian HP , whose groundstate encodes the solution. Although it is easy to construct HP , finding its ground state might becomputationally difficult. Thus we take HP = H(T ), which implies that |ψg(T )〉 encodes our solution.For intermediate times, H(t) smoothly interpolates between H(0) and H(T ) = HP . If time T is largeenough, H(t) will indeed be slowly varying and the final state |ψ(T )〉 will be close to the solution of theproblem encoded in |ψg(T )〉. The cruicial question is how to estimate the appropriate time T .

3.3.1 MINIMUM SPECTRAL GAP

Adiabatic quantum evolution of Hamiltonian can be generally expressed as

H(t) = [1− s(t)]Hinit + s(t)HP ,

with s(t) changing from 0 to 1.The performance of AQC is determined by the minimal gap

gm = min0≤s≤1

(E1(s)− E0(s)).

The adiabatic theorem imposes the minimum time it takes for the switching from Hinit to HP to beadiabatic. This time can be thought of as the algorithm complexity! If H(s) has an exponentially smallminimum gap (with respect to the number of qubits used in computation) then the corresponding algo-rithm is inefficient, whereas minimum gap which scales inverse polynomially gives an efficient quantumadiabatic algorithm whose running time is also polynomial.

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3.3.2 LOCAL VS. GLOBAL EVOLUTION

In the global adiabatic evolution scheme, s is changed uniformly with time (s = const.) and the compu-tation time scales as τglobal ∝ g−2

m . On the other hand, in the local adiabatic scheme s in a non-linearfunction of time chosen in such a way to optimize the computation time. 24 In this case τlocal ∝ g−1

m !

Global evolution After evolution under H(s) for a time T , the system is found in the ground stateof HP with probability (1− ε2)2, provided the evolution rate satisfies

|〈dHdt 〉1,0|g2min

≤ ε,⟨dHdt

⟩1,0

=⟨E1; t|dH

dt|E0; t

⟩, (1)

where ε � 1. The above formula (Eq. (1)) follows directly from adiabatic theorem by applying first-order perturbation theory on a two-level system of relevant states. In particular, Eq. (1) implies thatthe minimum gap cannot be smaller than a certain value if we require the state at time t to differ frominstantaneous ground state by a negligible amount (a smaller gap implies a higher transition probabilityto the first excited state). As long as the gap is finite, for any finite and positive ε, the time of evolutioncan be finite.

Local evolution Adiabatic evolution scheme can be improved, since we have applied Eq. (1) to theentire time interval T , hence imposing the limit on the evolution rate during the whole computationwhile this limit is only severe in the vicinity of gmin. Thus, by dividing T into infinitesimal time intervalsdt and applying adiabaticity condition locally to each of these intervals, we can vary the evolution ratecontinuously in time and thereby speeding up the computation. New condition would be

|dsdt| ≤ ε g2(t)

|〈dHds 〉1,0|, (2)

for all times t. 25

The local and global schemes of AQC are also different in their response to decoherence. The globalscheme is robust against environmental noise, on the contrary, local adiabatic scheme is very sensitiveto decoherence. It was shown that in order for the local scheme to change the scaling time from ∝ g−2

m

to ∝ g−1m , the computation time should be smaller than global dephasing time. 26 Beside that, local

adiabatic evolution requires knowledge of the spectrum which is not feasible for general Hamiltonians.

3.3.3 LANDAU-ZENER FORMULA

Landau-Zener formula [15] is an analytic solution to the equations of motions governing the transitiondynamics of a 2-level quantum mechanical system, with a time-dependent Hamiltonian varying suchthat the energy separation of the two states is a linear function of time.

If the system starts in the lower energy eigenstate we are interested in the probability of findingthe system in the upper energy eigenstate in infinite future (so called Landau-Zener transition). Forinfinitely slow variation of the energy difference, the adiabatic theorem tells us that no such transitionwill take place, as the system will always be in an instantaneous eigenstate of the Hamiltonian at some

24This is done by spending the majority of time in the vicinity of anticrossing.25For instance, local evolution applied to adiabatic Grover’s search algorithm provides quadratic speed up over global

evolution, enabling to solve the problem in total running time of order√N (same as logic gate quantum computer)[18].

26Characteristic time over which the mutual phases between qubits are destroyed. This implies loss of information fromthe system.

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time. At non-zero velocities, transitions (diabatic) occur with probability given by the Landau-Zenerformula

PD = e−2πΓ, Γ =a2/~

| ∂∂t (E2 − E1)|=

a2~

| (dqdt

)︸︷︷︸vLZ

∂∂q (E2 − E1)|

,

where q is the perturbation variable (e.g. electric or magnetic field), E1,2 are the energies of the crossingstates and vLZ denotes Landau-Zener velocity which is inversely proportional to time. The quantity ais the off-diagonal element of the two-level Hamiltonian coupling given eigenstates.

Since computational time is always finite, there is a nonzero probability that system would undergoLandau-Zener transitions and end up in an exited state. Hence, Landau-Zener transitions are importantin estimating the sufficient computation time.

4 WHAT KIND OF PROBLEMS CAN AQC SOLVE?AQC is not restricted only to optimization problems (unlike QA) - an universal AQC can run anyquantum algorithm, and has been shown to be computationally equivalent to the gate model of quatnumcomputation, as both can be efficiently mapped into each other [16].

Quantum adiabatic algorithms have been aplied to solve various optimization problems, for instancefinding cliques 27 in random graphs [17]. There is no know classical algorithm that finds the largestclique in a random graph with high probability and runs in polynomial time.

In these algorithms, the condition for adiabaticity is fulfilled globaly by using only the minimumenergy gap between ground and first exited state to determine the computation time. This method(global evolution) is not efficient in some cases, such as adiabatic Grover’s search algorithm [18] andadiabatic Deutsch-Jozsa algorithm [19] as they result in a complexity τglobal = O(N) (which is complexityof classical algorithms). However these algorithms can be improved by application of local adiabaticevolution yielding an optimal performance of a quantum algorithm, τlocal = O(

√N ).

On the other hand, the universal AQC can provide solution to a problem in polynomial time ifthe same problem can be solved in polynomial time using logic gate quantum computer. Evidently,the polynomial advantage does not depend on the local evolution (instead it only provides a quadraticenhancement).

4.1 RECENT DEVELOPEMENTAn important question is what kind of problems can benefit from AQC without requiring local adiabaticevolution and therefore phase coherence? Using a perturbative approach to estimate the gap size ofadiabatic quantum optimization it has been found that the gap is inversely proportional to the squareroot of the number of states that have energies close to global minimum, which means if the number oflow energy local minima becomes exponentially large, then the gap will be exponentially small. In suchcases, only a local adiabatic evolution scheme can provide advantage over classical computation. LocalAQC however requires phase coherence during the evolution and knowledge of the energy spectrum,which limits its practicality [20].

These problems, although unsuitable for AQC, could still be suitable for heuristic algorithm (suchis quantum annealing) if approximate solutions are acceptable, because the chance of finding a solutionwithin the acceptance range will be large.

Equally important question is whether the interactions between the computer and its environmentcan spoil the computation. It is clear that AQC has fundamental advantages over the gate model in

27A clique in an undirected graph is a subset of its vertices such that every two vertices are connected by an edge.

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regards to robustness against decoherence, however there does not yet exist an equivalent of the thresholdtheorem [5] that describes under what conditions AQC coupled to an environment will succeed [21].

In an isolated system with no decoherence the limitation is due to the usual Landau-Zener tunnelingat the anticrossing. Probability to leave out of the ground state is given by the a adiabatic theorem.The main assumption here is that there exist a well-defined energy gap between the two lowest energystates. In reality, the energy levels of qubit register are broadened by the coupling to an environment.

Figure 5: Broadening of the energy levels of a closed system (a) due to coupling to the environmentmade of a single two-state system (b) or infinitely many degrees of freedom with a continuous energyspectrum (c). In latter case the anticrossing turns into a continuous transition region [22].

Since the broadening W typically increases with the number of qubits (Fig. 5), while the minimumgap gm decreases, the realistic large-scale system will eventually fall in the (incoherent) regime W � gm[22].

5 PRACTICAL IMPLEMENTATION OF ADIABATIC QUAN-TUM COMPUTER

First experimental implementation of Shor’s algorithm was demonstrated by Vandersypen et. al. [23]in 2001 using nuclear spins to find the prime factors of number 15. More recent experiments by Lu et.al. [24] and Lanyon et. al. [25] used photons as qubits and found the same factors. In 2005, Mitraet. al demonstrated the experimental implementations of local adiabatic evolution algorithms (Grover’ssearch and Deutsch-Jozsa algorithm) on a 2-qubit quantum information processor using NMR [26, 27].

Chuang et. al. [28] have demonstrated the implementation of a quantum adiabatic algorithm bysolving MAXCUT 28 problem on a 3-qubit system by NMR.

Note that in actual implementation, the Hamiltonian of the system is discretized in order to recastadiabatic evolution in terms of unitary operators,

U =M∏m=0

Um, Um = exp (−i[(1− m

M)Hinit +

m

MHP ]∆t), ∆t = T/(M + 1).

Discretizing a continuous Hamiltonian is straightforward process and changes the total run time T28One wants a subset S of vertex set such that the number of edges between S and complementary subset is as large as

possible.

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of the adiabatic evolution only polynomially. The required adiabatic limit is achieved when both T andnumber of discrete steps M approach infinity.

5.1 D-WAVE SYSTEMS (The Quantum Computing Company)D-Wave’s core focus is the developement of superconducting processors capable of running adiabaticquantum algorithms for solving quadratic unconstrained binary optimization problems (NP-hard op-timization problem) 29. Many important scientific and commercial problems require the solution ofQUBO.

D-Wave processors actually perform quantum annealing. Combinatorial optimization is representedby a disordered Ising spin Hamiltonian, e.g.

H0(t) =∑i=1

hiSzi +

N∑i,j

JijSzi S

zj +

∑i

∆i(t)Syi

Example is taken from a Hydra processor manufactured by D-Wave. Si are representing Paulimatrices, hi is the local bias on qubit i, Jij is the coupling strength between qubits i and j and ∆i(t)tunneling matrix element. A problem instance is encoded in the h and J values. The traverse term isused to control the quantum annealing schedule [29].

Many artificial intelligence problems can be mapped to NP-hard optimization problems, particularyQUBO is found to be very useful in pattern matching, common in machine learning applications [30].

D-Wave processors are fabricated using superconducting metals instead of semiconductors and areoperated at ultra-low temperatures.

A circuit consisting of a network of coupled compound Josephson junction rf-SQUID flux qubits hasbeen used to impelement an adiabatic quantum optimization algorithm [31].

Flux qubit Flux qubits (also known as persistent current qubits, are micro-metre sized loops ofsuperconducting metal interrupted by a number of Josephson junctions. The junction parameters areengineered during fabrication so that a persistent current will flow continuously when an external flux isapplied. The computational basis states of the qubit are defined by the circulating currents which canflow either clockwise or counter-clockwise. These currents screen the applied flux limiting it to multiplesof the flux quantum and give the qubit its name. When the applied flux through the loop area is closeto a half integer number of flux quanta the two energy levels corresponding to the two directions ofcirculating current are brought close together and the loop may be operated as a qubit.

Computational operations are performed by pulsing the qubit with microwave frequency radiationwhich has an energy comparable to that of the gap between the energy of the two basis states.

29QUBO is given by the formula E(X1, . . . , XN ) =PN

i≤j QijXiXj , Xi ∈ {0, 1}

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Figure 6: Large loop interrupted by two Josephson junctions (the SQUID) merged with the smallerloop on the right side comprising three in-line Josephson junctions (the flux qubit). Arrows indicate thedirection of the persistent current for each qubit state and the corresponding measured Rabi oscillations(cyclic behaviour of a two-state quantum system in the presence of an oscillatory driving field) are shownbelow. Image is taken from qsd.magnet.fsu.edu.

ConclusionsWe have seen that the efficiency of AQC approach is fundamentally limited by the small spectral gapsbetween ground and exited states. It was shown that these gaps can become exponentially small underspecific conditions, such as bad choice of initial Hamiltonian [32] or specifically designed hard instances.As it unfortunately seems, exponentially small gaps appear close to the end of the adiabatic algorithmfor large random instances of NPC problems, which indicates the failure of the adiabatic quantum opti-mization (the sistem gets trapped in one of the numerous local minima) [33, 36]. However it is importantto point out that we are still lacking the rigorous analytical result characterizing the performance of AQCon random instances of NPC problems. Anyway, by assuming worst case scenario AQC could still befound more efficient in comparisson to classical algorithms on average case.

References[1] Deutsch D.: Quantum theory, the Church-Turing principle and the universal quantum computer

Proceedings of the Royal Society of London A 400, pp. 97-117 (1985)

[2] Petek A.: Kvantna logična vrata, http://mafija.fmf.uni-lj.si/seminar/seminar.php, seminar(2008)

[3] Gregorič M.: Kvantni računalniki, http://mafija.fmf.uni-lj.si/seminar/seminar.php, semi-nar (2007)

[4] List of qubit prepresentations: http://en.wikipedia.org/wiki/Qubit, (2010)

[5] Nielsen M.A., Chuang I.L.: Quantum computation & quantum information, Cambridge UniversityPress (2000)

[6] Raussendorf R., Briegel H.J: A One-Way Quantum Computer, Phys. Rev. Lett. 86 (2001)

[7] Raussendorf R., Harrington J., Goyal K.: A fault-tolerant one-way quantum computer, Annals ofPhysics 321 (2006)

[8] Quantum Dot Cellular Automata: http://en.wikipedia.org/wiki/Quantum_dot_cellular_automata (19.12.2009)

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[9] Cellular Automata: http://en.wikipedia.org/wiki/Cellular_automata (19.12.2009)

[10] Braid theory: http://en.wikipedia.org/wiki/Braid_theory (19.12.2009)

[11] Topological Quantum Computer: http://en.wikipedia.org/wiki/Topological_quantum_computer (19.12.2009)

[12] Goldman V.J., Camino F.E., Zhou W.: Realization of a Lauhlin Quasiparticle Interferometer: Ob-servation of Fractional Statistics, Phys. Rev. Lett. B 72 (2005)

[13] List I.: Kvantni topološki računalniki, http://mafija.fmf.uni-lj.si/seminar/seminar.php,seminar (2009)

[14] Quantum annealing: http://en.wikipedia.org/wiki/Quantum_annealing, (2010)

[15] Landau-Zener formula: http://en.wikipedia.org/wiki/Landau-Zener_formula (20.12.2009)

[16] Aharonov D., van Dam W., Kempe J., Landau Z., Lyold S., Regev O.: Adiabatic Quantum Com-putation is Equivalent to Standard Quantum Computation, SIAM Journal of Computing, Vol. 37,Issue 1 (2007)

[17] Childs A.M., Farhi E., Goldstone J., Gutmann S.: Finding cliques by quantum adiabatic evolution,Quantum Information and Computation 2, 181 (2002)

[18] Roland J., Cerf J. N.: Quantum Search by Local Adiabatic Evolution, Phys. Rev. A 65 (2002)

[19] Das S., Kobes R., Kunstatter G.: Adiabatic Quantum Computation and Deutsch’s Algorithm, Phys.Rev. A65 (2002)

[20] Amin M. H. S.: Effect of Local Minima od Adiabatic Quantum Optomization, Phys. Rev. Lett. 100(2008)

[21] Johansson J., Amin M.H.S., Berkley A.J., Bunyk P., Choi V., Harris R., Johnson M.W., Lant-ing T.M., Lloyd S., Rose G.: Landau-Zener Transitions in an Adiabatic Quantum Computer,arXiv:0807.0797v1 (2008)

[22] Amin M.H.S., Averin D.V., Nesteroff J.A.: Decoherence in Adiabatic Quantum Computation,arXiv:0708.0384v3 (2009)

[23] Vandersypen M.K.L, Steffen M., Breyta G., Yannoni C.S., Sherwood M.H., Chuang I.L.: Experi-mental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance, Nature414 (2001)

[24] Lu C.Y., Browne D.E., Yang T., Pan J.W.: Demonstration of Shor’s quantum factoring algorithmusing photonic qubits, Phys. Rev. Lett. 99 (2007)

[25] Lanyon B.P, Weinhold T.J., Langford N.K., Barbieri M., James D.F.V., Gilchrist A., White A.G.:Experimental demonstration of Shor’s algorithm with quantum entanglement, Phys. Rev. Lett. 99(2007)

[26] Mitra A., Ghosh A., Das R., Patel A. Kumar A.: Experimental implementation of local adiabaticevolution algorithms by an NMR quantum information processor, arXiv:quant-ph/0503060v2 (2005)

[27] Tulsi A., Mitra A., Kumar A.: Experimental NMR implementation of a robust quantum searchalgorithm, arXiv:0912.4071v1 (2009)

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[28] Chuang I.L., Steffen M., van Dam W., Hogg T., Breyta G.: Experimental implementation of anadiabatic quantum optimization algorithm, Phys. Rev. Lett. 90 (2003)

[29] Rose G., Macready W.G.: An Introduction to Quantum Annealing, D-Wave Systems (2007)

[30] Neven H., Rose G., Macready W.G.: Image recognition with an adiabatic quantum computer I.Mapping to quadratic unconstrained binary optimization, arXiv:0804.4457v1 (2008)

[31] Harris R., Berkley A.J., Johansson J. et.al.: Implementation of a Quantum Annealing AlgorithmUsing a Superconducting Circuit

[32] Žnidarič M., Horvat M.: Exponential complexity of an adiabatic algorithm for an NP-completeproblem

[33] Altschuler B., Krovi H., Roland J.: Anderson localization casts clouds over adiabatic quantumoptimization, arXiv:0912.0746v1 (2009)

[34] Farhi E., Goldstone J., Gutmann S., Sipser M.: Quantum Computation by Adiabatic Evolution,arXiv:quant-ph/0001106v1 (2000)

[35] Farhi E., Goldstone J., Gutmann S., Lapan J., Lundgren A., Preda D. A Quantum AdiabaticEvolution Algorithm Applied to Random Instances of an NP-Complete Problem, arXiv:quant-ph/0104129v1 (2001)

[36] Altschuler B., Krovi H., Roland J.: Anderson localization cast clouds over adiabatic quantum opti-mization, arXiv:quant-ph/0912.0746v1 (2009)

[37] Farhi E., Goldstone J., Gosset D., Gutmann S., Meyer B. H., Shor P.: Quantum Adiabatic Algo-rithms, Small Gaps, and Different Paths, arXiv:0909.4766v1 [quant-ph] (2009)

Other internet sources (2010):http://en.wikipedia.org/wiki/Adiabatic_quantum_computationhttp://en.wikipedia.org/wiki/Quantum_computerhttp://en.wikipedia.org/wiki/Quantum_decoherencehttp://en.wikipedia.org/wiki/Adiabatic_theoremhttp://en.wikipedia.org/wiki/One-way_quantum_computerhttp://en.wikipedia.org/wiki/Flux_qubithttp://info.phys.unm.edu/~alandahl/iqiwiki/index.php/Cluster_State_Quantum_Computing

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