Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXP QMA-EXP and is hard for FEXP NEXP . This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FP QMA-EXP and is hard for FP NEXP . Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground energies of the resulting matrices than were required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction that shows hardness for a function class.

We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class $FP^{NEXP}$. Gottesman and Irani (FOCS 2009) also studied classical translationally-invariant constraint satisfaction problems; they show that the problem of deciding whether the cost of the optimal solution is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel's famous result from 1988, showing that the function version of SAT is complete for the class $FP^{NP}$. An essential ingredient in the proof is a study of the complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of $Ω(N^{1/4})$, for an $N \times N$ grid. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for $P^{NEXP}$.

We give a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit in the regime of anti-concentration to within inverse polynomial total variation distance. The algorithm is based on a quantum analog of noise induced low degree approximations of Boolean functions, which takes the form of the truncation of a Feynman path integral in the Pauli basis.

Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXPQMA-EXP and is hard for FEXPNEXP. This means that the problem is doubly exponentially hard in the size of the input.

Despite the interest in the complexity class MA, the randomized analog of NP, just a few natural MA-complete problems are known. The first problem was found by (Bravyi and Terhal, SIAM Journal of Computing 2009); it was then followed by (Crosson, Bacon and Brown, PRE 2010) and (Bravyi, Quantum Information and Computation 2015). Surprisingly, two of these problems are defined using terminology from quantum computation, while the third is inspired by quantum computation and keeps a physical terminology. This prevents classical complexity theorists from studying these problems, delaying potential progress, e.g., on the NP vs. MA question. Here, we define two new combinatorial problems and prove their MA-completeness. The first problem, ACAC, gets as input a succinctly described graph, with some marked vertices. The problem is to decide whether there is a connected component with only unmarked vertices, or the graph is far from having this property. The second problem, SetCSP, generalizes standard constraint satisfaction problem (CSP) into constraints involving sets of strings. Technically, our proof that SetCSP is MA-complete is based on an observation by (Aharonov and Grilo, FOCS 2019), in which it was noted that a restricted case of Bravyi and Terhal's problem (namely, the uniform case) is already MA-complete; a simple trick allows to state this restricted case using combinatorial language. The fact that the first, more natural, problem of ACAC is MA-hard follows quite naturally from this proof, while the containment of ACAC in MA is based on the theory of random walks. We notice that the main result of Aharonov and Grilo carries over to the SetCSP problem in a straightforward way, implying that finding a gap-amplification procedure for SetCSP (as in Dinur's PCP proof) is equivalent to MA=NP. This provides an alternative new path towards the major problem of derandomizing MA.

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant ε, is in NP. Thus, if there exists a gap-amplification procedure for uniform stoquastic Local Hamiltonians (in analogy to the gap amplification procedure for constraint satisfaction problems in the original PCP theorem), then MA = NP (and vice versa). Furthermore, if this gap amplification procedure exhibits some additional (natural) properties, then P = RP. We feel this work opens up a rich set of new directions to explore, which might lead to progress on both quantum PCP and derandomization. We also provide two small side results of potential interest. First, we are able to generalize our result by showing that deciding if a uniform stoquastic Local Hamiltonian has negligible or constant frustration can be also solved in NP. Additionally, our work reveals a new MA-complete problem which we call SetCSP, stated in terms of classical constraints on strings of bits, which we define in the appendix. As far as we know this is the first (arguably) natural MA-complete problem stated in non-quantum CSP language.

Analog quantum simulations---simulations of one Hamiltonian by another---is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian $\tilde{H}$ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian $H$ (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where $\tilde{H}$ is only required to simulate the groundstate(s) and spectral gap of $H$ instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves a good starting point for future research of these topics.

Mochon's proof [Quantum Weak Coin Flipping with Arbitrarily Small Bias, preprint, arXiv:0711.4114, 2007] of the existence of quantum weak coin flipping with arbitrarily small bias is a fundamental result in quantum cryptography, but at the same time one of the least understood. Though used several times as a black box in important follow-up results [M. Ganz, Quantum Leader Election, preprint, arXiv:0910.4952, 2009; A. Chailloux and I. Kerenidis, in Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 527--533; N. Aharon and J. Silman, New J. Phys., 12 (2010), 033027; A. Chailloux and I. Kerenidis, in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2011; I. Kerenidis and S. Zhang, in Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science 7582, Springer, Berlin, 2013, pp. 13--28], the result has not been peer reviewed, its novel techniques (and, in particular, Kitaev's point game formalism) have not been applied anywhere else, and an explicit protocol is missing. We believe that truly understanding the existence proof and the novel techniques it relies on would constitute a major step in quantum information theory, leading to deeper understanding of entanglement and of quantum protocols in general. In this work, we make a first step in this direction. We simplify parts of Mochon's construction considerably, making about $20$ pages of analysis in the original proof superfluous, clarifying some other parts of the proof on the way, and presenting the proof in a way which is conceptually easier to grasp. We believe the resulting proof of existence is easier to understand, more readable, and certainly verifiable. Moreover, we analyze the resources neededto achieve a bias $\varepsilon$ and show that the number of qubits is $O(\log \frac{1}{\varepsilon})$, while the number of rounds is $(\frac{1}{\varepsilon})^{O(\frac{1}{\varepsilon})}$. A true understanding of the proof, including Kitaev's point-game techniques and their applicability, as well as completing the task of constructing an explicit (and also simpler and more efficient) protocol, are left to future work.

We initiate the study of quantum locally testable codes ($\text{qLTC}$s). Classical $\text{LTC}$s are very important in computational complexity. These codes are defined as the linear subspace satisfying a set of local constraints, with the additional requirement that their soundness, $R(\delta)$, which is the probability that a randomly chosen constraint is violated, is proportional to the proximity $\delta$, where $\delta n$ is the distance of a word from the code. Excellent $\text{LTC}$s exist in the classical world, and they are tightly related to the celebrated $\text{PCP}$ (probabilistically checkable proof) theorem. In quantum complexity, quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement, and they have played a crucial role in many computational complexity results. We provide a definition of the quantum analogue of $\text{LTC}$s and motivate it by connecting its central notions in the study of both entanglement and quantum Hamiltonian complexity. A natural question is whether such codes exist, and how good can their soundness be. To the best of our knowledge all quantum codes known today exhibit poor soundness. Moreover, we show that the soundness of $\text{CSS}$ codes (which are commonly used quantum codes defined by two classical codes) is governed by the minimal soundness of the two classical codes; in the most natural $\text{CSS}$ code we examined as a candidate $\text{qLTC}$, namely, the Reed--Muller code, there is a tradeoff between the parameters of the two codes, which prevents the resulting quantum code from being $\text{qLTC}$. These facts seem to suggest a more general phenomenon, by which the soundness of $\text{qLTC}$s is inherently restricted due to multiparticle entanglement. Our main technical contribution consists of two complementary results regarding $\text{qLTC}$s which are stabilizer codes (denoted $\text{sLTC}$s). We first prove a surprising, inherently quantum property of $\text{sLTC}$s. For small constant values of proximity, the better the local expansion of the interaction graph of the constraints, the less sound the $\text{sLTC}$ becomes. This stands in sharp contrast to the classical setting. The complementary, more intuitive result also holds (and is actually much more involved technically to prove in the quantum case): an upper bound on the soundness when the code is defined on bad local expanders. Together we arrive at a quantum upper bound on the soundness of $\text{sLTC}$s set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for $\text{qLTC}$s, and their relation to other objects of interest in quantum information theory. In the appendix we also define a quantum analogue of $\text{PCP}$s of proximity ($\text{PCPP}$s) and point out that the result of [E. Ben-Sasson et al., SIAM J. Comput., 36 (2006), pp. 889--974] by which $\text{PCPP}$s imply $\text{LTC}$s with related parameters carries over to the $\text{sLTC}$s. This creates a first link between $\text{qLTC}$s and quantum $\text{PCP}$s [D. Aharonov, I. Arad, and T. Vidick, ACM SIGACT News Archive, 44 (2013), pp. 47--79].

We introduce a technique for applying quantum expanders in a distributed fashion, and use it to solve two basic questions: testing whether a bipartite quantum state shared by two parties is the maximally entangled state and disproving a generalized area law. In the process these two questions which appear completely unrelated turn out to be two sides of the same coin. Strikingly in both cases a constant amount of resources are used to verify a global property.

The classical description of quantum states is in general exponential in the number of qubits. Can we get polynomial descriptions for more restricted sets of states such as ground states of interesting subclasses of local Hamiltonians? This is the basic problem in the study of the complexity of ground states, and requires an understanding of multi-particle entanglement and quantum correlations in such states. Area laws provide a fundamental ingredient in the study of the complexity of ground states, since they offer a way to bound in a quantitative way the entanglement in such states. Although they have long been conjectured for many body systems in arbitrary dimensions, a general rigorous was only recently proved in Hastings' seminal paper [8] for ID systems. In this paper, we give a combinatorial proof of the ID area law for the special case of frustration free systems, improving by an exponential factor the scaling in terms of the inverse spectral gap and the dimensionality of the particles. The scaling in terms of the dimension of the particles is a potentially important issue in the context of resolving the 2D case and higher dimensions, which is one of the most important open questions in Hamiltonian complexity. Our proof is based on a reformulation of the detectability lemma, introduced by us in the context of quantum gap amplification [1]. We give an alternative proof of the detectability lemma, which is not only simpler and more intuitive than the original proof, but also removes a key restriction in the original statement, making it more suitable for this new context. We also give a one page proof of Hastings' proof that the correlations in the ground states of gapped Hamiltonians decay exponentially with the distance, demonstrating the simplicity of the combinatorial approach for those problems.

The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003 [10], using a clever application of the representation theory of C*-algebras, that if the terms in the Hamiltonian are all two-local, the problem is in NP, and the entanglement in the ground states is local. The general case remained open since then. In this paper we extend this result beyond the two-local case, to the case of three-qubit interactions. We then extend our results even further, and show that NP verification is possible for three-wise interaction between qutrits as well, as long as the interaction graph is planar and also "nearly Euclidean" in some well-defined sense. The proofs imply that in all such systems, the entanglement in the ground states is local. These extensions imply an intriguing sharp transition phenomenon in commuting Hamiltonian systems: the ground spaces of 3-local "physical" systems based on qubits and qutrits are diagonalizable by a basis whose entanglement is highly local, while even slightly more involved interactions (the particle dimensionality or the locality of the interaction is larger) already exhibit an important long-range entanglement property called Topological Order. Our results thus imply that Kitaev's celebrated Toric code construction is, in a well defined sense, optimal as a construction of Topological Order based on commuting Hamiltonians.

The quantum analog of a constraint satisfaction problem is a sum of local Hamiltonians- each (term of the) Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. Formalizing the intuitive connection between the ground (minimal) energy of the Hamiltonian and the minimum number of violated constraints is problematic, since the number of constraints being violated is not well defined when the terms in the Hamiltonian do not commute. The detectability lemma proved in this paper provides precisely such a quantitative connection. We apply the lemma to derive a quantum analogue of the classical gap amplification lemma of random walks on expander graphs. The quantum gap amplification lemma holds for local Hamiltonians with expander interaction graphs. Our proofs are based on a novel structure imposed on the Hilbert space, which we call the XY decomposition, which enables a reduction from the quantum non-commuting case to the commuting case (where many classical arguments go through). The results may have several interesting implications. First, proving a quantum analogue to the PCP theorem is one of the most important challenges in quantum complexity theory. Our quantum

This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when $\eta$, the probability for an error in a qubit or a gate, is smaller than a constant threshold $\eta_c$. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate $\eta$ decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with $p>2$ states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be $\eta_c\simeq 10^{-6}$, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum' particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P.The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP ne QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two‐dimensional grid with nearest neighbor interactions. The equivalence between the models allows stating the main open problems in quantum computation using well‐studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices.

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to this task by studying the problem of quantum state generation. We motivate this problem by showing that the entire class of statistical zero knowledge, which contains natural candidates for efficient quantum algorithms such as graph isomorphism and lattice problems, can be reduced to the problem of quantum state generation. To study quantum state generation, we define a paradigm which we call adiabatic state generation (ASG) and which is based on adiabatic quantum computation. The ASG paradigm is not meant to replace the standard quantum circuit model or to improve on it in terms of computational complexity. Rather, our goal is to provide a natural theoretical framework, in which quantum state generation algorithms could be designed. The new paradigm seems interesting due to its intriguing links to a variety of different areas: the analysis of spectral gaps and ground‐states of Hamiltonians in physics, rapidly mixing Markov chains, adiabatic computation, and approximate counting. To initiate the study of ASG, we prove several general lemmas that can serve as tools when using this paradigm. We demonstrate the application of the paradigm by using it to turn a variety of (classical) approximate counting algorithms into efficient quantum state generators of nontrivial quantum states, including, for example, the uniform superposition over all perfect matchings in a bipartite graph.

The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13, 14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e2π i/5, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13, 14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2π i/k, where the running time of the algorithm is polynomial in m,n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem.The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards self-reducible P-hard problems, most notably, the Potts model.

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere---we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev[2003]. An interesting fact is that our result emerged from a “dequantization” of our previous quantum result in Aharonov and Regev [2003]. This route to proving purely classical results might be beneficial elsewhere.

The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of /spl radic/n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk (1993), Goldreich and Goldwasser (2000), and Aharonov and Regev (2003). Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier transform over the lattice. This technique might be useful elsewhere - we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP,) improving on a previous result of Regev (2003). An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in (Aharanov and Regev, 2003). This route to proving purely classical results might be beneficial elsewhere.

We consider coGapSVP/sub /spl radic/n/, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM /spl cap/ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'.We first show that any problem in Statistical Zero Knowledge (including eg. discrete log, quadratic residuosity and gap closest vector in a lattice) can be reduced to an instance of the quantum state generation problem. Having shown the generality of the state generation problem, we set the foundations for a new paradigm for quantum state generation. We define 'Adiabatic State Generation' (ASG), which is based on Hamiltonians instead of unitary gates. We develop tools for ASG including a very general method for implementing Hamiltonians (The sparse Hamiltonian lemma), and ways to guarantee non negligible spectral gaps (The jagged adiabatic path lemma). We also prove that ASG is equivalent in power to state generation in the standard quantum model. After setting the foundations for ASG, we show how to apply our techniques to generate interesting superpositions related to Markov chains.The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks. We hope that these links will bring new insights and methods into quantum algorithms.

We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

Article Free Access Share on Quantum bit escrow Authors: Dorit Aharonov University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Amnon Ta-Shma University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Umesh V. Vazirani University of California, Berkeley, CA University of California, Berkeley, CAView Profile , Andrew C. Yao Princeton University, Princeton, NJ Princeton University, Princeton, NJView Profile Authors Info & Claims STOC '00: Proceedings of the thirty-second annual ACM symposium on Theory of computingMay 2000 Pages 705–714https://doi.org/10.1145/335305.335404Published:01 May 2000Publication History 46citation619DownloadsMetricsTotal Citations46Total Downloads619Last 12 Months56Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Current formal models for quantum computation deal only with unitary gates operating on "pure quantum states". In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subroutines, and more. It turns out, that the restriction to unitary gates and pure states is unnecessary. In this paper we generalize the formal model of quantum circuits to a model in which the state can be a general quantum state, namely a mixed state, or a "density matrix", and the gates can be general quantum operations, not necessarily unitary. The new model is shown to be equivalent in computational power to the standard one, and the problems mentioned above essentially disappear. The main result in this paper is a solution for the subroutine problem. The general function that a quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density matrices, we prove a simple lower bound on depth of circuits that compute probabilistic functions. Finally, we deal with the question of inaccurate quantum computation with mixed states. Using the so called "trace metric" on density matrices, we show how to keep track of errors in the new model. Institutes of Physics and Computer science, The Hebrew University, Jerusalem, Israel y L.D.Landau Institute for Theoretical Physics,Moscow, Russia z Institute of Compu...

In the past year many developments have taken place in the area of quantum error corrections. Recently Shor showed how to perform fault tolerant quantum computation when, , the probability for a fault in one time step per qubit or per gate, is polylogarithmically small. This paper closes the gap and shows how to perform fault tolerant quantum computation when the error probability, , is smaller than some constant threshold, 0 . The cost is polylogarithmic in time and space, and no measurements are used during the quantum computation. The same result is shown also for quantum circuits which operate on nearest neighbors only.

Recently it has become clear, that a key issue in quantum computation is understanding how interaction with the environment, or "decoherence", affects the computational power of quantum computers. We adopt the standard physical method of describing systems which are interwound with their environment by "density matrices", and within this framework define a model of decoherence in quantum computation. Our results show that the computational power of decohered quantum computers depends strongly on the amount of parallelism in the computation. We first present a simulation of decohered sequential quantum computers, on a classical probabilistic Turing machine, and prove that the expected slowdown of this simulation is polynomial in time and space of the quantum computation, for any non zero decoherence rate. Similar results hold for quantum computers that are allowed to operate on logarithmic number of qubits at a time. For decohered quantum circuits (with local gates), the situation is more subtle and depends on the decoherence rate, /spl eta/. We find that our simulation is efficient for circuits with decoherence rate /spl eta/ higher than some constant /spl eta//sub 1/ but exponential for a general (random) circuit subjected to decoherence rate lower than some constant /spl eta//sub 2/. The transition from exponential cost to polynomial cost happens in a short range of decoherence rates. We use computer experiments to exhibit the phase transitions in various quantum circuits.