arXiv subjects

Sevag Gharibian

Publications and source records attributed to Sevag Gharibian.

At least 37 records · Page 2Linked to original sources

arXiv2023

The 7 faces of quantum NP

When it comes to NP, its natural definition, its wide applicability across scientific disciplines, and its timeless relevance, the writing is on the wall: There can be only one. Quantum NP, on the other hand, is clearly the apple that fell far from the tree of NP. Two decades since the first definitions of quantum NP started rolling in, quantum complexity theorists face a stark reality: There's QMA, QCMA, QMA1, QMA(2), StoqMA, and NQP. In this article aimed at a general theoretical computer science audience, I survey these various definitions of quantum NP, their strengths and weaknesses, and why most of them, for better or worse, actually appear to fit naturally into the complexity zoo.

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arXiv2023

Optimizing the depth of variational quantum algorithms is strongly QCMA-hard to approximate

Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the \emph{depth} of the variational ``ansatz'' used -- the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant $ε>0$, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor $N^{1-ε}$, for $N$ denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even ``simpler'' QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.

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arXiv2022

Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement

Savitch's theorem states that NPSPACE computations can be simulated in PSPACE. We initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), where an exponentially long classical proof is streamed to a poly-space quantum verifier. Besides two main results, we also show that a quantum analogue of Savitch's theorem is unlikely to hold, as SQCMASPACE=NEXP. For completeness, we introduce Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE=QMA_EXP (quantum analogue of NEXP). Our first main result shows, in contrast to the classical setting, the solution space of a quantum constraint satisfaction problem (i.e. a local Hamiltonian) is always connected when exponentially long proofs are permitted. For this, we show how to simulate any Lipschitz continuous path on the unit hypersphere via a sequence of local unitary gates, at the expense of blowing up the circuit size. This shows quantum error-correcting codes can be unable to detect one codeword erroneously evolving to another if the evolution happens sufficiently slowly, and answers an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State Connectivity problem. Our second main result is that any SQCMASPACE computation can be embedded into "unentanglement", i.e. into a quantum constraint satisfaction problem with unentangled provers. Formally, we show how to embed SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux, Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of scaling the promise gap with the streamed proof size. As a corollary, we obtain the first systematic construction for obtaining QMA(2)-type upper bounds on arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap scales exponentially with the number of bits of communication in the interactive proof.

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arXiv2021

On polynomially many queries to NP or QMA oracles

We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$ and $P^{QMA}$, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes $P^{NP[\log]}$ and $P^{QMA[\log]}$, defined identically to $P^{NP}$ and $P^{QMA}$, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a $P^{NP}$ machine have a "query graph" which is a tree, then this computation can be simulated in $P^{NP[\log]}$. In this work, we first show that for any verification class $C\in\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}\}$, any $P^C$ machine with a query graph of "separator number" $s$ can be simulated using deterministic time $\exp(s\log n)$ and $s\log n$ queries to a $C$-oracle. When $s\in O(1)$ (which includes the case of $O(1)$-treewidth, and thus also of trees), this gives an upper bound of $P^{C[\log]}$, and when $s\in O(\log^k(n))$, this yields bound $QP^{C[\log^{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding $P^C$ computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial $p$ specified via an arithmetic circuit, if one can "weakly compress" $p$ so that its optimal value requires $m$ bits to represent, then $P^{NP}$ can be decided with only $m$ queries to an NP-oracle.

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arXiv2021

Towards Quantum One-Time Memories from Stateless Hardware

A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings. Here, we propose a scheme for using quantum information, together with the assumption of stateless (i.e., reusable) hardware tokens, to build statistically secure OTMs. Via the semidefinite programming-based quantum games framework of Gutoski and Watrous [STOC 2007], we prove security for a malicious receiver making at most 0.114n adaptive queries to the token (for n the key size), in the quantum universal composability framework, but leave open the question of security against a polynomial amount of queries. Compared to alternative schemes derived from the literature on quantum money, our scheme is technologically simple since it is of the "prepare-and-measure" type. We also give two impossibility results showing certain assumptions in our scheme cannot be relaxed.

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arXiv2021

On efficiently solvable cases of Quantum k-SAT

The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k>=3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

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arXiv2020

The Complexity of Translationally Invariant Problems beyond Ground State Energies

It is known that three fundamental questions regarding local Hamiltonians -- approximating the ground state energy (the Local Hamiltonian problem), simulating local measurements on the ground space (APX-SIM), and deciding if the low energy space has an energy barrier (GSCON) -- are $\mathsf{QMA}$-hard, $\mathsf{P}^{\mathsf{QMA}[log]}$-hard and $\mathsf{QCMA}$-hard, respectively, meaning they are likely intractable even on a quantum computer. Yet while hardness for the Local Hamiltonian problem is known to hold even for translationally-invariant systems, it is not yet known whether APX-SIM and GSCON remain hard in such "simple" systems. In this work, we show that the translationally invariant versions of both APX-SIM and GSCON remain intractable, namely are $\mathsf{P}^{\mathsf{QMA}_{\mathsf{EXP}}}$- and $\mathsf{QCMA}_{\mathsf{EXP}}$-complete, respectively. Each of these results is attained by giving a respective generic "lifting theorem" for producing hardness results. For APX-SIM, for example, we give a framework for "lifting" any abstract local circuit-to-Hamiltonian mapping $H$ (satisfying mild assumptions) to hardness of APX-SIM on the family of Hamiltonians produced by $H$, while preserving the structural and geometric properties of $H$ (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we "compress" the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we give a hardness construction robust against highly non-local unitaries, i.e. even if the adversary acts on all but one qudit in the system in each step.

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arXiv2020

The complexity of simulating local measurements on quantum systems

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following lower and upper bounds. Lower bounds (hardness results): (1) The P^QMA[log]-completeness result of [Ambainis, CCC 2014] requires O(log n)-local observables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. (2) We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete. (3) We identify a flaw in [Ambainis, CCC 2014] regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on [Ambainis, CCC 2014] to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to show P^QMA[log] is in PP. This improves the containment QMA is in PP [Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a promise problem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.

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arXiv2019

Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

The polynomial-time hierarchy ($\mathrm{PH}$) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as $\mathrm{PH}$ does not collapse). Here, we study whether two quantum generalizations of $\mathrm{PH}$ can similarly prove separations in the quantum setting. The first generalization, $\mathrm{QCPH}$, uses classical proofs, and the second, $\mathrm{QPH}$, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, $\mathrm{Q} Σ_3$, into $\mathrm{NEXP}$ {using the Ellipsoid Method for efficiently solving semidefinite programs}. These results yield two implications for $\mathrm{QMA}(2)$, the variant of Quantum Merlin-Arthur ($\mathrm{QMA}$) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if $\mathrm{QCPH} = \mathrm{QPH}$ (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then $\mathrm{QMA}(2)$ is in the Counting Hierarchy (specifically, in $\mathrm{P}^{\mathrm{PP}^{\mathrm{PP}}}$). Second, unless $\mathrm{QMA}(2)={\mathrm{Q} Σ_3}$ (i.e., alternating quantifiers do not help in the presence of "unentanglement"), $\mathrm{QMA}(2)$ is strictly contained in $\mathrm{NEXP}$.

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arXiv2019

Almost optimal classical approximation algorithms for a quantum generalization of Max-Cut

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-ferromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively.

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arXiv2019

Oracle complexity classes and local measurements on physical Hamiltonians

The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted to more physical Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. We first give a sequence of results which together yield P^QMA[log]-hardness for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. These equalities simplify the proofs of our subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians, including physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.

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arXiv2019

Tensor network non-zero testing

Tensor networks are a central tool in condensed matter physics. In this paper, we study the task of tensor network non-zero testing (TNZ): Given a tensor network T, does T represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic k-SAT problem on D-dimensional qudits is in NP for logarithmic k and constant D. This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic k, (2) constant D, and (3) for arbitrary interaction graphs.

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arXiv2019

Ground state connectivity of local Hamiltonians

The study of ground state energies of local Hamiltonians has played a fundamental role in quantum complexity theory. In this paper, we take a new direction by introducing the physically motivated notion of "ground state connectivity" of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories. Roughly, "ground state connectivity" corresponds to the natural question: Given two ground states |ψ> and |ϕ> of a local Hamiltonian H, is there an "energy barrier" (with respect to H) along any sequence of local operations mapping |ψ> to |ϕ>? We show that the complexity of this question can range from QCMA-complete to PSPACE-complete, as well as NEXP-complete for an appropriately defined "succinct" version of the problem. As a result, we obtain a natural QCMA-complete problem, a goal which has generally proven difficult since the conception of QCMA over a decade ago. Our proofs rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions. We show that this lemma is essentially tight with respect to the length of the unitary evolution in question.

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arXiv2018

Quantum One-Time Memories from Stateless Hardware

A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings. Here, we show how to use quantum information, together with the assumption of stateless (i.e., reusable) hardware tokens, to build statistically secure OTMs. This is in sharp contrast with the classical case, where stateless hardware tokens alone cannot yield OTMs. In addition, our scheme is technologically simple. We prove security in the quantum universal composability framework, employing semi-definite programming results of Molina, Vidick and Watrous [TQC 2013] and combinatorial techniques of Pastawski et al. [Proc. Natl. Acad. Sci. 2012].

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arXiv2016

Quantum Hamiltonian Complexity

Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.

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arXiv2015

A linear time algorithm for quantum 2-SAT

The Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979].

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arXiv2013

Quantifying non-classicality with local unitary operations

We propose a measure of non-classical correlations in bipartite quantum states based on local unitary operations. We prove the measure is non-zero if and only if the quantum discord is non-zero; this is achieved via a new characterization of zero discord states in terms of the state's correlation matrix. Moreover, our scheme can be extended to ensure the same relationship holds even with a generalized version of quantum discord in which higher-rank projective measurements are allowed. We next derive a closed form expression for our scheme in the cases of Werner states and (2 x N)-dimensional systems. The latter reveals that for (2 x N)-dimensional states, our measure reduces to the geometric discord [Dakic et al., PRL 105, 2010]. A connection to the CHSH inequality is shown. We close with a characterization of all maximally non-classical, yet separable, (2 x N)-dimensional states of rank at most two (with respect to our measure).

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arXiv2013

Gate-efficient discrete simulations of continuous-time quantum query algorithms

We show how to efficiently simulate continuous-time quantum query algorithms that run in time T in a manner that preserves the query complexity (within a polylogarithmic factor) while also incurring a small overhead cost in the total number of gates between queries. By small overhead, we mean T within a factor that is polylogarithmic in terms of T and a cost measure that reflects the cost of computing the driving Hamiltonian. This permits any continuous-time quantum algorithm based on an efficiently computable driving Hamiltonian to be converted into a gate-efficient algorithm with similar running time.

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