arXiv subjects

Umesh Vazirani

Publications and source records attributed to Umesh Vazirani.

At least 37 records · Page 2Linked to original sources

arXiv2017

Rigorous RG algorithms and area laws for low energy eigenstates in 1D

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance of Landau et al. gave a polynomial time algorithm to actually compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly($n$) dimensional), or for the poly($n$) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified RG type transformation. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly($n$) degenerate ground spaces and an $n^{O(\log n)}$ algorithm for the poly($n$) lowest energy states for 1D systems (under a mild density condition). We note that for these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is $\tilde{O}(nM(n))$, where $M(n)$ is the time required to multiply two $n\times n$ matrices.

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arXiv2014

Local tests of global entanglement and a counterexample to the generalized area law

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.

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arXiv2014

How "Quantum" is the D-Wave Machine?

Recently there has been intense interest in claims about the performance of the D-Wave machine. In this paper, we outline a simple classical model, and show that it achieves excellent correlation with published input-output behavior of the D-Wave One machine on 108 qubits. While raising questions about "how quantum" the D-Wave machine is, the new model also provides additional algorithmic insights into the nature of the native computational problem solved by the D-Wave machine.

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arXiv2014

Comment on "Distinguishing Classical and Quantum Models for the D-Wave Device"

The SSSV model is a simple classical model that achieves excellent correlation with published experimental data on the D-Wave machine's behavior on random instances of its native problem, thus raising questions about how "quantum" the D-Wave machine is at large scales. In response, a recent preprint by Vinci et al. proposes a particular set of instances on which the D-Wave machine behaves differently from the SSSV model. In this short note, we explain how a simple modeling of systematic errors in the machine allows the SSSV model to reproduce the behavior reported in the experiments of Vinci et al.

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arXiv2013

A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians

Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including recently discovered structural features of gapped 1D systems, convex programming, insights from classical algorithms for 1D satisfiability, and new techniques for manipulating and bounding the complexity of MPS. Our result provides one of the first major classes of Hamiltonians for which computing ground states is provably tractable despite the exponential nature of the objects involved.

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arXiv2013

An area law and sub-exponential algorithm for 1D systems

We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$. Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).

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arXiv2012

Fully device independent quantum key distribution

The laws of quantum mechanics allow unconditionally secure key distribution protocols. Nevertheless, security proofs of traditional quantum key distribution (QKD) protocols rely on a crucial assumption, the trustworthiness of the quantum devices used in the protocol. In device-independent QKD, even this last assumption is relaxed: the devices used in the protocol may have been adversarially prepared, and there is no a priori guarantee that they perform according to specification. Proving security in this setting had been a central open problem in quantum cryptography. We give the first device-independent proof of security of a protocol for quantum key distribution that guarantees the extraction of a linear amount of key even when the devices are subject to a constant rate of noise. Our only assumptions are that the laboratories in which each party holds his or her own device are spatially isolated, and that both devices, as well as the eavesdropper, are bound by the laws of quantum mechanics. All previous proofs of security relied either on the use of many independent pairs of devices, or on the absence of noise.

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arXiv2012

A classical leash for a quantum system: Command of quantum systems via rigidity of CHSH games

Can a classical system command a general adversarial quantum system to realize arbitrary quantum dynamics? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. Here we report a technique by which a classical system can certify the joint, entangled state of a bipartite quantum system, as well as command the application of specific operators on each subsystem. This is accomplished by showing a strong converse to Tsirelson's optimality result for the Clauser-Horne-Shimony-Holt (CHSH) game: the only way to win many games is if the bipartite state is close to the tensor product of EPR states, and the measurements are the optimal CHSH measurements on successive qubits. This leads directly to a scheme for device-independent quantum key distribution. Control over the state and operators can also be leveraged to create more elaborate protocols for realizing general quantum circuits, and to establish that QMIP = MIP*.

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arXiv2012

Classical command of quantum systems via rigidity of CHSH games

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arXiv2012

Multiplicative Updates in Coordination Games and the Theory of Evolution

We study the population genetics of Evolution in the important special case of weak selection, in which all fitness values are assumed to be close to one another. We show that in this regime natural selection is tantamount to the multiplicative updates game dynamics in a coordination game between genes. Importantly, the utility maximized in this game, as well as the amount by which each allele is boosted, is precisely the allele's mixability, or average fitness, a quantity recently proposed in [1] as a novel concept that is crucial in understanding natural selection under sex, thus providing a rigorous demonstration of that insight. We also prove that the equilibria in two-person coordination games can have large supports, and thus genetic diversity does not suffer much at equilibrium. Establishing large supports involves answering through a novel technique the following question: what is the probability that for a random square matrix A both systems Ax = 1 and A^T y = 1 have positive solutions? Both the question and the technique may be of broader interest. [1] A. Livnat, C. Papadimitriou, J. Dushoff, and M.W. Feldman. A mixability theory for the role of sex in evolution. Proceedings of the National Academy of Sciences, 105(50):19803-19808, 2008.

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arXiv2012

Is Quantum Mechanics Falsifiable? A computational perspective on the foundations of Quantum Mechanics

Quantum computation teaches us that quantum mechanics exhibits exponential complexity. We argue that the standard scientific paradigm of "predict and verify" cannot be applied to testing quantum mechanics in this limit of high complexity. We describe how QM can be tested in this regime by extending the usual scientific paradigm to include {\it interactive experiments}.

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arXiv2012

An improved 1D area law for frustration-free systems

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings' 1D area law, and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $ε>0$ and interaction strength of at most $J$, our entropy bound is $S_{1D}\le \orderof{1}X^3\log^8 X$ where $X\EqDef(J\log d)/ε$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. Incorporating locality into the proof when applied to the 2D case gives an entanglement bound that is at the cusp of being non-trivial in the sense that any further improvement would yield a sub-volume law.

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arXiv2011

Quantum Hamiltonian complexity and the detectability lemma

Quantum Hamiltonian complexity studies computational complexity aspects of local Hamiltonians and ground states; these questions can be viewed as generalizations of classical computational complexity problems related to local constraint satisfaction (such as SAT), with the additional ingredient of multi-particle entanglement. This additional ingredient of course makes generalizations of celebrated theorems such as the PCP theorem from classical to the quantum domain highly non-trivial; it also raises entirely new questions such as bounds on entanglement and correlations in ground states, and in particular area laws. We propose a simple combinatorial tool that helps to handle such questions: it is a simplified, yet more general version of the detectability lemma introduced by us in the more restricted context on quantum gap amplification a year ago. Here, we argue that this lemma is applicable in much more general contexts. We use it to provide a simplified and more combinatorial proof of Hastings' 1D area law, together with a less than 1 page proof of the decay of correlations in gapped local Hamiltonian systems in any constant dimension. We explain how the detectability lemma can replace the Lieb-Robinson bound in various other contexts, and argue that it constitutes a basic tool for the study of local Hamiltonians and their ground states in relation to various questions in quantum Hamiltonian complexity.

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arXiv2008

The Detectability Lemma and Quantum Gap Amplification

The quantum analogue of a constraint satisfaction problem is a sum of local Hamiltonians - each local 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 a basic primitive in classical complexity: amplification of probabilities by random walks on expander graphs. It holds under the restriction that the interaction graph of the local Hamiltonian is an expander. Our proofs are based on a novel structure imposed on the Hilbert space that 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 gap amplification lemma may be viewed as the quantum analogue of the first of the three main steps in Dinur's PCP proof. Quantum gap amplification may also be related to spectral gap amplification, and in particular, to fault tolerance of adiabatic computation. Finally, the detectability lemma, and the $XY$ decomposition provide a handle on the structure of local Hamiltonians and their ground states.

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arXiv2007

A classical one-way function to confound quantum adversaries

The promise of quantum computation and its consequences for complexity-theoretic cryptography motivates an immediate search for cryptosystems which can be implemented with current technology, but which remain secure even in the presence of quantum computers. Inspired by recent negative results pertaining to the nonabelian hidden subgroup problem, we present here a classical algebraic function $f_V(M)$ of a matrix $M$ which we believe is a one-way function secure against quantum attacks. Specifically, inverting $f_V$ reduces naturally to solving a hidden subgroup problem over the general linear group (which is at least as hard as the hidden subgroup problem over the symmetric group). We also demonstrate a reduction from Graph Isomorphism to the problem of inverting $f_V$; unlike Graph Isomorphism, however, the function $f_V$ is random self-reducible and therefore uniformly hard. These results suggest that, unlike Shor's algorithm for the discrete logarithm--which is, so far, the only successful quantum attack on a classical one-way function--quantum attacks based on the hidden subgroup problem are unlikely to work. We also show that reconstructing any entry of $M$, or the trace of $M$, with nonnegligible advantage is essentially as hard as inverting $f_V$. Finally, $f_V$ can be efficiently computed and the number of output bits is less than $1+ε$ times the number of input bits for any $ε> 0$.

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arXiv2002

How Powerful is Adiabatic Quantum Computation?

We analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'.

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arXiv2002

Quantum Walks On Graphs

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.

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arXiv2000

Quantum Bit Escrow

Unconditionally secure bit commitment and coin flipping are known to be impossible in the classical world. Bit commitment is known to be impossible also in the quantum world. We introduce a related new primitive - {\em quantum bit escrow}. In this primitive Alice commits to a bit $b$ to Bob. The commitment is {\em binding} in the sense that if Alice is asked to reveal the bit, Alice can not bias her commitment without having a good probability of being detected cheating. The commitment is {\em sealing} in the sense that if Bob learns information about the encoded bit, then if later on he is asked to prove he was playing honestly, he is detected cheating with a good probability. Rigorously proving the correctness of quantum cryptographic protocols has proved to be a difficult task. We develop techniques to prove quantitative statements about the binding and sealing properties of the quantum bit escrow protocol. A related primitive we construct is a quantum biased coin flipping protocol where no player can control the game, i.e., even an all-powerful cheating player must lose with some constant probability, which stands in sharp contrast to the classical world where such protocols are impossible.

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