Dorit Aharonov

STOC Conference 2023 Conference Paper

A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling

• Dorit Aharonov
• Xun Gao
• Zeph Landau
• Yunchao Liu 0002
• Umesh V. Vazirani

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.

STOC Conference 2022 Conference Paper

Hamiltonian complexity in the thermodynamic limit

• Dorit Aharonov
• Sandy Irani

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 states of the resulting matrices than was 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 which shows hardness for a function class.

FOCS Conference 2019 Conference Paper

Stoquastic PCP vs. Randomness

• Dorit Aharonov
• Alex Bredariol Grilo

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.

FOCS Conference 2014 Conference Paper

Local Tests of Global Entanglement and a Counterexample to the Generalized Area Law

• Dorit Aharonov
• Aram W. Harrow
• Zeph Landau
• Daniel Nagaj
• Mario Szegedy
• Umesh V. Vazirani

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.

FOCS Conference 2011 Conference Paper

On the Complexity of Commuting Local Hamiltonians, and Tight Conditions for Topological Order in Such Systems

• Dorit Aharonov
• Lior Eldar

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.

FOCS Conference 2011 Conference Paper

The 1D Area Law and the Complexity of Quantum States: A Combinatorial Approach

• Dorit Aharonov
• Itai Arad
• Zeph Landau
• Umesh V. Vazirani

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.

STOC Conference 2009 Conference Paper

The detectability lemma and quantum gap amplification

• Dorit Aharonov
• Itai Arad
• Zeph Landau
• Umesh V. Vazirani

The quantum analogue of the constraint satisfaction problem is the fundamental physics question of finding the minimal energy state of a local Hamiltonian --- each term of the Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. However, in general it is not meaningful to ask for the probability that a given quantum state violates at least one constraint; the difficulty being that the terms of the Hamiltonian do not commute. We show how to make sense of this notion under mild restrictions on the form of the Hamiltonian. We then provide two main results. We first prove the quantum detectability lemma , which states that the probability of detecting a violation of a constraint in a local Hamiltonian system is bounded from below by some constant times the minimal energy of the system. The proof reveals some intrinsic structure of the Hilbert space of local Hamiltonians, which is captured in the "exponential decay" lemma, and formalized using a novel decomposition of the Hilbert space called the XY decomposition. As an application of the detectability lemma, we prove our second main result: a quantum analogue of the classical gap amplification lemma using random walks over expander graphs, which was the seed for Dinur's celebrated new proof of the PCP theorem [6]. We hope that these results will pave the way to better understandings of the computational properties of local Hamiltonians systems, and to the evolving field of quantum Hamiltonian complexity.

FOCS Conference 2007 Conference Paper

The Power of Quantum Systems on a Line

• Dorit Aharonov
• Daniel Gottesman
• Sandy Irani
• Julia Kempe

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.

STOC Conference 2006 Conference Paper

A polynomial quantum algorithm for approximating the Jones polynomial

• Dorit Aharonov
• Vaughan Jones
• Zeph Landau

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, e 2 π 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 e 2π 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.

FOCS Conference 2004 Conference Paper

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

• Dorit Aharonov
• Wim van Dam
• Julia Kempe
• Zeph Landau
• Seth Lloyd
• Oded Regev 0001

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.

FOCS Conference 2004 Conference Paper

Lattice Problems in NP cap coNP

• Dorit Aharonov
• Oded Regev 0001

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.

FOCS Conference 2003 Conference Paper

A Lattice Problem in Quantum NP

• Dorit Aharonov
• Oded Regev 0001

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.

STOC Conference 2003 Conference Paper

Adiabatic quantum state generation and statistical zero knowledge

• Dorit Aharonov
• Amnon Ta-Shma

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.

STOC Conference 2001 Conference Paper

Quantum walks on graphs

• Dorit Aharonov
• Andris Ambainis
• Julia Kempe
• Umesh V. Vazirani

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.

STOC Conference 2000 Conference Paper

Quantum bit escrow

• Dorit Aharonov
• Amnon Ta-Shma
• Umesh V. Vazirani
• Andrew Chi-Chih Yao

STOC Conference 1998 Conference Paper

Quantum Circuits with Mixed States

• Dorit Aharonov
• Alexei Y. Kitaev
• Noam Nisan

STOC Conference 1997 Conference Paper

Fault-Tolerant Quantum Computation With Constant Error

• Dorit Aharonov
• Michael Ben-Or

FOCS Conference 1996 Conference Paper

Polynomial Simulations of Decohered Quantum Computers

• Dorit Aharonov
• Michael Ben-Or

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.