A measurement driven analog of adiabatic quantum computation for frustration-free Hamiltonians
Liming Zhao, Carlos A. Perez-Delgado, Simon C. Benjamin, Joseph F. Fitzsimons
aa r X i v : . [ qu a n t - ph ] J un A measurement driven analog of adiabatic quantum computation for frustration-free Hamiltonians
Liming Zhao,
1, 2
Carlos A. P´erez-Delgado, Simon C. Benjamin, and Joseph F. Fitzsimons
1, 2, ∗ Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 School of Computing, University of Kent, Canterbury CT2 7NF, United Kingdom Department of Materials, University of Oxford, Parks Rd, Oxford OX1 3PH, United Kingdom
The adiabatic quantum algorithm has drawn intense interest as a potential approach to accelerating opti-mization tasks using quantum computation. The algorithm is most naturally realised in systems which supportHamiltonian evolution, rather than discrete gates. We explore an alternative approach in which slowly varyingmeasurements are used to mimic adiabatic evolution. We show that for certain Hamiltonians, which remainfrustration-free all along the adiabatic path, the necessary measurements can be implemented through the mea-surement of random terms from the Hamiltonian. This offers a new, and potentially more viable, method ofrealising adiabatic evolution in gate-based quantum computer architectures.
In the field of quantum computation, it has long been recog-nized that there exists deep connections between ground statesof Hamiltonians and problems of fundamental interest to thestudy of computational complexity [1, 2]. It is known that theproblem of finding the ground state of a Hamiltonian is hardeven in the case of one-dimensional lattices [3], and that ingeneral the k -local Hamiltonian problem is QMA-hard (andhence NP-hard) for any k ≥ = NP, such algorithms often work for Hamiltoniansof practical interest.One such quantum algorithm is the adiabatic algorithm[13], which is fundamentally rooted in the adiabatic theorem[14]. Informally, the adiabatic theorem states that a systemstarting in the ground state of some initial Hamiltonian willstay close to the ground state of the system if the Hamilto-nian is gradually changed over time, provided that this changeis continuous and sufficiently slow. This means that one canprepare the ground state of an arbitrary Hamiltonian H f byfirst preparing the ground state of some simple Hamiltonian H I and then subjecting the system to a time varying Hamil-tonian which slowly interpolates between H I and H F . Inits simplest form, the adiabatic algorithm considers a linearinterpolation between the initial and final Hamiltonians de-scribed by H ( s ) = ( − s ) H I + s H F for s ∈ [ , ] , where s is some simple function of time. This provides a heuristic ap-proach for tackling satisfiability problems [13, 15]. In general,the timescale required for this evolution can be exponentiallylong, as it scales with the reciprocal of the gap between theground state and first excited state of the instantaneous Hamil-tonians at each point in time. This reconciles the adiabatic ap-proach with the fact that QMA is not known to be contained inBQP, the class of problems efficiently solvable on a quantumcomputer. Indeed, it is now known that the adiabatic model isequivalent to circuit model quantum computation [16].Due to its wide applicability as a black-box optimizationtechnique, the adiabatic algorithm and similar techniques such ∗ Electronic address: joseph fi[email protected] as quantum annealing have emerged as one of the key use-cases for quantum processors [17]. The efficient implementa-tion of such techniques raises architectural concerns, however.While adiabatic evolution is in principle possible in manymonolithic quantum processor architectures, the Hamiltonianspossible are often restricted to 2-local interactions accordingto some fixed graph [18]. While techniques have been devisedto overcome these limitations, they incur significant overhead[19–21]. The situation is far worse when one considers thecase of distributed quantum computing architectures, such asmany promising ion-trap and quantum dot proposals [22, 23],which implement entangling operations between nodes usingdiscrete operations rather than Hamiltonian dynamics. Forsuch systems, a direct implementation of adiabatic computa-tion requires simulating Hamiltonian dynamics with discretelogic gates, an approach which would incur prohibitive over-head [24].Here we show that it is possible to implement adiabatic-likeevolution using relatively simple measurements provided thatthe Hamiltonian remains frustration free at all points alongthe adiabtic path. Our results are based on a connection be-tween the adiabatic theorem and the quantum Zeno effect[25]. We begin by presenting an alternate proof of a resultdue to Somma, Boixo and Knill which gave an adiabatic-liketheorem for systems measured (or dephased) in the eigen-bases of slowly varying Hamiltonians. We then show that forfrustration-free Hamiltonians, measurement of randomly cho-sen individual terms of the Hamiltonian suffices to approxi-mate measurement of the ground state, satisfying our crite-rion for adiabatic-like evolution. For k -local Hamiltonians,these measurement have constant complexity, as they corre-spond to projectors on at most k qubits. This potentially opensthe door to a direct analogue of the adiabatic algorithm wellsuited for distributed architectures, such as ion-trap imple-mentations and similar systems currently under investigation[26, 27]. These results also provide some level of theoreti-cal understanding of the mechanism behind a measurement-driven approach to SAT-solving proposed by Benjamin [28]which has shown promising performance in numerical exper-iments.We start by considering the evolution of the state of a quan-tum system due to the measurement of a sequence of observ-ables, which we treat as corresponding directly to Hamiltoni-ans. We then prove that, provided the difference between pairsof neighbouring Hamiltonians in the sequence has sufficientlysmall norm compared to the energy gap between the groundstate and first excited state, a system prepared in the groundstate of the initial Hamiltonian will evolve to the ground stateof the final Hamiltonian with high probability.Let H I and H F be the initial and final Hamiltonian respec-tively. Also, let { H n } ≤ n ≤ N be an ordered set of intermediateinterpolating operators, such that H ≡ H I and H N ≡ H F .For simplicity, we will assume that every H n is normalizedsuch that the eigenvalues lie in the range between 0 and 1, withthe lowest eigenvalue being exactly 0. The assumption on therange of the eigenvalues can be made without loss of general-ity, as the Hamiltonians can always be rescaled by multiplyingby a constant and shifted by adding a multiple of the identity.We will make no assumption regarding the degeneracy of theground state space. Taking | ψ i to be a state in the groundstate space of H I , and taking | ψ n i to denote the normalizedprojection of | ψ n − i onto the ground state space of H n , theevolution of the system then satisfies the following constraint. Theorem 1.
Given a system initially in state | ψ i , the state | ψ N i can be obtained with probability p ≥ − ε by measuringthe operators H n in sequence for ≤ n ≤ N, provided that max ≤ n ≤ N k ∆ H n k ∞ g ( H n ) ! ≤ ε N , where g ( H n ) is the gap between the eigenvalues correspond-ing to the ground state space and first excited state of H n ,and ∆ H n = H n − H n − . Furthermore, if at each step n themeasurement of H n is replaced with any procedure that pro-duces a state ρ n , such that the trace distance from | ψ n ih ψ n | is at most δ N , with probability at least h ψ n | ρ n − | ψ n i , thenthe overall procedure yields a state ρ N , with trace distance atmost δ N from | ψ N ih ψ N | , with probability p ′ ≥ − ε − δ .Proof. Taking P n to be the projector onto the ground statespace of H n , then the probability of successfully obtaining | ψ n i from | ψ n − i is given by p n = k P n | ψ n − ik . Then, theprobability of successfully projecting onto | ψ N i during the fi-nal measurement is bounded by p ≥ ∏ n p n . The reason thisis a bound rather than an exact equality is due to the possi-bility of reaching the correct final state through a sequence ofmeasurements fails to project onto the ground state of someintermediate Hamiltonian.Now, consider the probability of failure at step n , assumingthat all previous measurements have successfully projectedonto the ground state space of the associated Hamiltonian, ε n = k ( I − P n ) | ψ n − ik . This can be turned into an inequality by making use ofLoewner order, noting that ( I − P n ) ≤ H n g ( H n ) I , and hence ε n ≤ (cid:13)(cid:13)(cid:13)(cid:13) H n g ( H n ) | ψ n − i (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) ∆ H n g ( H n ) | ψ n − i (cid:13)(cid:13)(cid:13)(cid:13) , where the equality follows from the fact that H n − | ψ n − i = p n . By making use of the definitionof the infinity norm for matrices, we arrive at p n ≥ − k ∆ H n k ∞ g ( H n ) . The final success probability is then bounded by p ≥ − N ∑ n = k ∆ H n k ∞ g ( H n ) . Provided that Eq. 1 holds, we then have p ≥ − ε as required.When considering the modified procedure, the modifiedprobability of success at each step is bounded from belowby p ′ n ≥ Tr ( P n ρ n − ) . This can be rewritten as p ′ n = p n + Tr ( P n ( ρ n − − | ψ n − ih ψ n − | )) . Using the trace distance con-straint, this implies p ′ n ≥ − ε N − δ N and hence p ′ ≥ − ε − δ as required.While Eq. 1 may appear unusual when compared to adia-batic conditions, due to the way in which N appears as a re-ciprocal it can be transformed into a more conventional formby making the substitution δ N H n = N ∆ H n , to obtain N ≥ ε − max ≤ n ≤ N k δ N H n k ∞ g ( H n ) ! . Suppose that for any N each of the measured Hamiltoni-ans H n is chosen along a fixed continuous path H ( s ) , for0 ≤ s ≤
1, through the space of Hamiltonians, such that theylie sequentially along this path at equal intervals. In this case,for large N the finite difference δ N H n tends to the deriva-tive dds H ( s ) , and is thus approximately constant for large N ,depending only on the path through the space of Hamiltoni-ans. Note that N does not have dimensions of time, and sothis equation is not directly comparable to adiabatic theorems.However, making the substitution T = N / max ≤ n ≤ N δ n H n one obtains a more conventional adiabatic expression (simi-lar to that in Ref. [29]).While the result presented above provides a link betweenthe measurement of interpolating Hamiltonians and the adi-abatic theorem, this does not imply that measurements are aviable alternative to Hamiltonian evolution for implementingadiabatic quantum computation. After all, the measurementof a Hamiltonian is a non-trivial task, and implementing it viacontrolled unitary evolution and phase estimation [30] mayprovide little advantage over directly implementing adiabaticevolution. In order to increase the utility of this correspon-dence, we now introduce a method for efficiently projectingonto the ground state of frustration-free Hamiltonians.Let H be a frustration-free Hamiltonian which is the sumof m terms, H = ∑ mi = ω i H i , where every term H i is a tensor product of 2 × ∑ mi = ω i = H i one canconstruct a POVM measurement with measurement operators E i = √ I − H i and ˜ E i = √ H i . Specifically, if the eigenvaluesare either 0 or 1, one can construct a projective measurementwith projectors H i and I − H i . The lowest energy subspace isobtained when the measurement result is I − H i .Now, consider the following operation M on an arbitraryquantum state ρ . First, an index 1 ≤ i ≤ m is selected at ran-dom with probability ω i . A POVM measurement is then per-formed on ρ with measurement operators E i and ˜ E i . If theoutcome of the measurement corresponds to application of ˜ E i then the procedure is said to fail. Otherwise, the resultingstate of the system is ρ ′ i = E i ρ E † i Tr ( E i ρ E † i ) . This latter case occurswith probability p ( s | i ) = Tr ( E i ρ E † i ) . Disregarding the choiceof i , the output state ρ ′ of a successful application of M willbe a mixed state consisting of a distribution over the variouspossibilities for ρ ′ i as follows. Let p ( s ) be the total successprobability. Since every i is chosen with probability p ( i ) = ω i ,we then have p ( s ) = m ∑ i = ω i Tr (cid:16) E i ρ E † i (cid:17) = Tr m ∑ i = ω i ( I − H i ) ρ ! = − Tr ( H ρ ) . (1)From Bayes’ theorem, the output state ρ ′ is then given by ρ ′ = m ∑ i = p ( i ) p ( s | i ) p ( s ) ρ ′ i = − Tr ( H ρ ) m ∑ i = ω i E i ρ E † i . We now show that successful application of the operation M to a state ρ , with non-zero overlap with the ground state space,will increase the projection onto the ground state space. Lemma 1.
Let H be a frustration-free Hamiltonian, as de-scribed above. Let P gs be the projector onto the ground statespace of H . Let ρ be an arbitrary density matrix and let ρ ′ be the resulting density matrix after a successful applicationof the operation M as defined above to ρ . Then, Tr (cid:0) P gs ρ ′ (cid:1) = Tr ( P gs ρ ) − Tr ( H ρ ) , (2) and the probability that M is successful is − Tr ( H ρ ) .Proof. We begin by noting thatTr (cid:0) P gs ρ ′ (cid:1) = − Tr ( H ρ ) Tr P gs m ∑ i = ω i E i ρ E † i ! . Using the cyclic property of trace, this can be rewritten asTr (cid:0) P gs ρ ′ (cid:1) = − Tr ( H ρ ) m ∑ i = ω i Tr (cid:16) E i P gs E † i ρ (cid:17) . (3)The measurement operators can then be absorbed into P gs .Evaluating the summation then yields Eq. 2 as required. Theprobability of success for applying M was previously calcu-lated in Eq. 1. We now consider what happens when M is applied not once,but some number of times k . Theorem 2.
Let H be a frustration-free Hamiltonian. LetP gs be the projector onto the ground state space of H . Let ρ be a density matrix with non-zero overlap with the groundstate space of H and let ρ ( k ) be the resulting density matrixafter a successful application of the operation M as definedabove to ρ sequentially k times. Then, Tr (cid:16) P gs ρ ( k ) (cid:17) ≥ (cid:18) + ( − g ( H )) k (cid:18) ( P gs ρ ) − (cid:19)(cid:19) − Furthermore, P gs ρ ( k ) P gs ∝ P gs ρ P gs and the probability that allk applications of M are successful is at least Tr ( P gs ρ ) .Proof. We will consider the ratio R ℓ = Tr (cid:16)(cid:0) I − P gs (cid:1) ρ ( ℓ ) (cid:17) Tr (cid:0) P gs ρ ( ℓ ) (cid:1) = Tr (cid:16) P gs ρ ( ℓ ) (cid:17) − − . (4)By definition ρ ( ℓ ) = M (cid:16) ρ ( ℓ − ) (cid:17) for all ℓ >
1, and hence fromLemma 1 it follows that R ℓ = (cid:16) − Tr (cid:16) H ρ ( ℓ − ) (cid:17)(cid:17) Tr (cid:16) P gs ρ ( ℓ − ) (cid:17) − − . Since Tr (cid:16) H ρ ( ℓ − ) (cid:17) ≥ g ( H ) (cid:16) − Tr (cid:16) P gs ρ ( ℓ − ) (cid:17)(cid:17) thisgives rise to the bound R ℓ ≤ ( − g ( H )) (cid:18) Tr (cid:16) P gs ρ ( ℓ − ) (cid:17) − − (cid:19) . Using Eq. 4 we then arrive at the recurrence inequality R ℓ ≤ ( − g ( H )) R ℓ − . Hence R k ≤ ( − g ( H )) k R . From Eq. 4 we can then replace R k and R to obtainTr (cid:16) P gs ρ ( k ) (cid:17) ≥ + ( − g ( H )) k (cid:0) P gs ρ (cid:1) − !! − as required.Turning to the projection of ρ ( k ) onto the ground statespace, from the definition of M we have ρ ( k ) = ∑ i ... i k E i k . . . E i ρ E † i . . . E † i k ∑ j ... j k Tr (cid:16) E j k . . . E j ρ E † j . . . E † j k (cid:17) and hence P gs ρ ( k ) P gs ∝ ∑ i ... i k P gs E i k . . . E i ρ E † i . . . E † i k P gs = ∑ i ... i k E i k . . . E i P gs ρ P gs E † i . . . E † i k = P gs ρ P gs . The success probability for applying M any number oftimes can be lower bounded by noting that M does not al-ter states in the ground state space of H . Hence the traceof the projection of ρ onto this subspace provides a lowerbound.Theorem 2 implies that applying M sufficiently many timessatisfies the requirements of Theorem 1 for a procedure ap-proximately projecting onto the ground state space of a Hamil-tonian. This can be made quantitative by noting that if k = α / g ( H ) then ( − g ( H )) k ≤ e − α . When used in thecontext of Theorem 1 it will necessarily be the case that (cid:18) ( P gs ρ ) − (cid:19) ≪
1. In such cases it should suffice to choose α ∝ log N to provide the necessary accuracy.The results presented above hold even for Hamiltonianswith degenerate ground states and thus are broadly applicable.The combination of these results provides a means for imple-menting adiabatic-like dynamics using measurements of onlymodest complexity, at least for frustration-free Hamiltonians. This suggests that such evolution can be realised without needfor Trotterisation of Hamiltonian dynamics, and provides apotentially more viable approach in quantum computers basedon discrete gates, especially in the context of distributed ar-chitectures. The restriction to frustration free Hamiltoniansis used to ensure that the ground state is simultaneously aneigenstate of each possible measurement. Removing this re-striction represents an interesting avenue for future research. Acknowledgments