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Dorit Aharonov

The Hebrew University of Jerusalem, School of Computer Science and Engineering, Faculty Member

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Papers by Dorit Aharonov

Stoquastic PCP vs. Randomness

arXiv (Cornell University), Jan 16, 2019

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In ... more 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 ǫ, 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.

Limitations of Noisy Reversible Computation

arXiv (Cornell University), Nov 17, 1996

In this paper we study noisy reversible circuits. Noisy computation and reversible computation ha... more In this paper we study noisy reversible circuits. Noisy computation and reversible computation have been studied separately, and it is known that they are equivalent in power to unrestricted computation. We study the case where both noise and reversibility are combined and show that the combined model is weaker than unrestricted computation. We consider the model of reversible computation with noise, where the value of each wire in the circuit is flipped with some fixed probability 1/2 > p > 0 each time step, and all the inputs to the circuit are present in time 0. We prove that any noisy reversible circuit must have size exponential in its depth in order to compute a function with high probability. This is tight as we show that any (not necessarily reversible or noise-resistant) circuit can be converted into a reversible one that is noise-resistant with a blow up in size which is exponential in the depth. This establishes that noisy reversible computation has the power of the complexity class N C 1. We extend the upper bound to quantum circuits, and prove that any noisy quantum circuit must have size exponential in its depth in order to compute a function with high probability. This highlight the fact that current error-correction schemes for quantum computation require constant inputs throughout the computation (and not just at time 0), and shows that this is unavoidable. As for the lower bound, we show that quasi-polynomial noisy quantum circuits are at least powerful as quantum circuits with logarithmic depth (or QN C 1). Making these bounds tight is left open in the quantum case.

A Simple Proof that Toffoli and Hadamard are Quantum Universal

arXiv (Cornell University), Jan 9, 2003

Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is... more Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a 'classical' set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev's universal set of gates [11], and discuss the meaning of the result.

Quantum NP - A Survey

arXiv (Cornell University), Oct 11, 2002

We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum ... more We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3−SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT , but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.

On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems

arXiv (Cornell University), Feb 3, 2011

The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Unde... more 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 [8], using clever applications 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 the results of Bravyi and Vyalyi 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 embedded on a planar lattice, or more generally, "Nearly Euclidean" (NE). 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: 3-local NE systems based on qubits and qutrits cannot be used to construct Topological order, as their entanglement is local, whereas for higher dimensional qudits, or for interactions of at least 4 qudits, Topological Order is already possible, via Kitaev's Toric Code construction. We thus conclude that Kitaev's Toric Code construction is optimal for deriving topological order based on commuting Hamiltonians.

Technical perspective: On proofs, entanglement, and games

Communications of The ACM, Oct 25, 2021

form of rigidity of quantum mechanics. It is as if the verifier gets to peek into the secret quan... more form of rigidity of quantum mechanics. It is as if the verifier gets to peek into the secret quantum labs of the provers! Moreover, a quantum version of low-degree tests enables selftests that verify exponentially long entangled states, using only polynomial communication. Now comes a mindboggling idea: though the verifier’s messages are too short to specify its exponentially long questions about the doubly exponentially long proof, it can use quantum self-testing to efficiently force the provers to share an exponentially large, entangled state and then to correctly sample their questions themselves. This result is just the start of the story; going all the way to the halting problem requires modifying the “compression-by-entanglement” idea so it can be applied recursively. A plethora of new hurdles arise, whose solution is the tour de force in this paper. The exciting applications include constructing infinite algebras that cannot be approximated in any finite dimension and separating quantum correlations models previously conjectured to be equal. Self-testing is connected to group stability, which raises hope for progress on major problems in group theory. How to explain the strong impact this theoretical computer science result has on pure mathematics? PCPs and other powerful computational complexity concepts are applied here, but perhaps another aspect is the role protocols play in the result. These sequences of individual steps that depend on time, constitute an intuitive way to think about highly complex mathematical objects, an approach that seems to offer a fresh look at physics and mathematical problems.

Quantum computation - alternative models and algorithms

On Quantum Advantage in Information Theoretic Single-Server PIR

Lecture Notes in Computer Science, 2019

In (single-server) Private Information Retrieval (PIR), a server holds a large database DB of siz... more In (single-server) Private Information Retrieval (PIR), a server holds a large database DB of size n, and a client holds an index i ∈ [n] and wishes to retrieve DB[i] without revealing i to the server. It is well known that information theoretic privacy even against an "honest but curious" server requires Ω(n) communication complexity. This is true even if quantum communication is allowed and is due to the ability of such an adversarial server to execute the protocol on a superposition of databases instead of on a specific database ("input purification attack"). Nevertheless, there have been some proposals of protocols that achieve sub-linear communication and appear to provide some notion of privacy. Most notably, a protocol due to Le Gall (ToC 2012) with communication complexity O( √ n), and a protocol by Kerenidis et al. (QIC 2016) with communication complexity O(log(n)), and O(n) shared entanglement. We show that, in a sense, input purification is the only potent adversarial strategy, and protocols such as the two protocols above are secure in a restricted variant of the quantum honest but curious (a.k.a specious) model. More explicitly, we propose a restricted privacy notion called anchored privacy, where the adversary is forced to execute on a classical database (i.e. the execution is anchored to a classical database). We show that for measurement-free protocols, anchored security against honest adversarial servers implies anchored privacy even against specious adversaries. Finally, we prove that even with (unlimited) pre-shared entanglement it is impossible to achieve security in the standard specious model with sub-linear communication, thus further substantiating the necessity of our relaxation. This lower bound may be of independent interest (in particular recalling that PIR is a special case of Fully Homomorphic Encryption).

Lattice Problems in NP &#8745; coNP

We show that the problems of approximating the shortest and closest vector in a lattice to within... more 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 [7], Goldreich and Goldwasser [14], and Aharonov and Regev [2]. 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 elsewherewe demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [26]. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in [2]. This route to proving purely classical results might be beneficial elsewhere.

SIGACT news, Jun 3, 2013

The classical PCP theorem is arguably the most important achievement of classical complexity theo... more The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.

A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

We give a polynomial time classical algorithm for sampling from the output distribution of a nois... more 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.

Translationally Invariant Constraint Optimization Problems

arXiv (Cornell University), Sep 18, 2022

We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically,... more We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the computational 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 constraint satisfaction problems using this strong notion of translational-invariance; they show that the problem of deciding whether the cost of the optimal assignment 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 computational 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 ), where the grid size is N × N . To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. This might be of independent interest. 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 .

On Gap-Simulation of Hamiltonians and the Impossibility of Quantum Degree-Reduction

arXiv (Cornell University), Apr 30, 2018

Analog quantum simulations-simulations of one Hamiltonian by another-is one of the major goals in... more 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 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 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 degreereduction 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.

Dining Philosophers, Leader Election and Ring Size problems, in the quantum setting

ArXiv, 2017

We provide the first quantum (exact) protocol for the Dining Philosophers problem (DP), a central... more We provide the first quantum (exact) protocol for the Dining Philosophers problem (DP), a central problem in distributed algorithms. It is well known that the problem cannot be solved exactly in the classical setting. We then use our DP protocol to provide a new quantum protocol for the tightly related problem of exact leader election (LE) on a ring, improving significantly in both time and memory complexity over the known LE protocol by Tani et. al. To do this, we show that in some sense the exact DP and exact LE problems are equivalent; interestingly, in the classical non-exact setting they are not. Hopefully, the results will lead to exact quantum protocols for other important distributed algorithmic questions; in particular, we discuss interesting connections to the ring size problem, as well as to a physically motivated question of breaking symmetry in 1D translationally invariant systems.

On Quantum Advantage in Information Theoretic Single-Server PIR

Advances in Cryptology – EUROCRYPT 2019, 2019

In (single-server) Private Information Retrieval (PIR), a server holds a large database DB of siz... more In (single-server) Private Information Retrieval (PIR), a server holds a large database DB of size n, and a client holds an index i ∈ [n] and wishes to retrieve DB[i] without revealing i to the server. It is well known that information theoretic privacy even against an "honest but curious" server requires Ω(n) communication complexity. This is true even if quantum communication is allowed and is due to the ability of such an adversarial server to execute the protocol on a superposition of databases instead of on a specific database ("input purification attack"). Nevertheless, there have been some proposals of protocols that achieve sub-linear communication and appear to provide some notion of privacy. Most notably, a protocol due to Le Gall (ToC 2012) with communication complexity O(√ n), and a protocol by Kerenidis et al. (QIC 2016) with communication complexity O(log(n)), and O(n) shared entanglement. We show that, in a sense, input purification is the only potent adversarial strategy, and protocols such as the two protocols above are secure in a restricted variant of the quantum honest but curious (a.k.a specious) model. More explicitly, we propose a restricted privacy notion called anchored privacy, where the adversary is forced to execute on a classical database (i.e. the execution is anchored to a classical database). We show that for measurement-free protocols, anchored security against honest adversarial servers implies anchored privacy even against specious adversaries. Finally, we prove that even with (unlimited) pre-shared entanglement it is impossible to achieve security in the standard specious model with sub-linear communication, thus further substantiating the necessity of our relaxation. This lower bound may be of independent interest (in particular recalling that PIR is a special case of Fully Homomorphic Encryption).

Quantum Computation

Annual Reviews of Computational Physics VI, Mar 1, 1999

In the last few years, theoretical study of quantum systems serving as computational devices has ... more In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. This question cannot be separated from that of quantum complexity, because any realistic model will inevitably be subject to such inaccuracies. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.

ABSTRACT Quantum Walks on Graphs

Lattice problems in NP n coNP

Jacm, 2005

Fault tolerant computation with constant error

Quantum Computation

In the last few years, theoretical study of quantum systems serving as computational devices has ... more In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of