arXiv subjects

Adam Bouland

Publications and source records attributed to Adam Bouland.

31 records · Page 2Linked to original sources

arXiv2018

Complexity Classification of Conjugated Clifford Circuits

Clifford circuits -- i.e. circuits composed of only CNOT, Hadamard, and $π/4$ phase gates -- play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) -- where one additionally conjugates every qubit by the same one-qubit gate $U$ -- can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary $U$ can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.

quant-ph↗

arXiv2018

Trading inverses for an irrep in the Solovay-Kitaev theorem

The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error $ε$ using merely $\mathrm{polylog}(1/ε)$ gates from any finite universal quantum gate set $\mathcal{G}$. One drawback to the theorem is that it requires the gate set $\mathcal{G}$ to be closed under inversion. Here we show that this restriction can be traded for the assumption that $\mathcal{G}$ contains an irreducible representation of any finite group $G$. This extends recent work of Sardharwalla et al. [arXiv:1602.07963], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [arXiv:quant-ph/0505030, arXiv:0908.0512].

quant-ph↗

arXiv2018

Quantum Supremacy and the Complexity of Random Circuit Sampling

A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS satisfies an anti-concentration property, making it the first supremacy proposal with both average-case hardness and anti-concentration.

quant-ph↗

arXiv2017

On the Power of Statistical Zero Knowledge

We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002 [Aar]. Second, we "lift" this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles relative to which SZK is not contained in PZK, NISZK is not contained in NIPZK, and PZK is not equal to coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity. The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.

cs.CC↗

arXiv2017

Rescuing Complementarity With Little Drama

The AMPS paradox challenges black hole complementarity by apparently constructing a way for an observer to bring information from the outside of the black hole into its interior if there is no drama at its horizon, making manifest a violation of monogamy of entanglement. We propose a new resolution to the paradox: this violation cannot be explicitly checked by an infalling observer in the finite proper time they have to live after crossing the horizon. Our resolution depends on a weak relaxation of the no-drama condition (we call it "little drama") which is the "complementarity dual" of scrambling of information on the stretched horizon. When translated to the description of the black hole interior, this implies that the fine-grained quantum information of infalling matter is rapidly diffused across the entire interior while classical observables and coarse-grained geometry remain unaffected. Under the assumption that information has diffused throughout the interior, we consider the difficulty of the information-theoretic task that an observer must perform after crossing the event horizon of a Schwarzschild black hole in order to verify a violation of monogamy of entanglement. We find that the time required to complete a necessary subroutine of this task, namely the decoding of Bell pairs from the interior and the late radiation, takes longer than the maximum amount of time that an observer can spend inside the black hole before hitting the singularity. Therefore, an infalling observer cannot observe monogamy violation before encountering the singularity.

hep-th↗

arXiv2016

Grover search and the no-signaling principle

Two of the key properties of quantum physics are the no-signaling principle and the Grover search lower bound. That is, despite admitting stronger-than-classical correlations, quantum mechanics does not imply superluminal signaling, and despite a form of exponential parallelism, quantum mechanics does not imply polynomial-time brute force solution of NP-complete problems. Here, we investigate the degree to which these two properties are connected. We examine four classes of deviations from quantum mechanics, for which we draw inspiration from the literature on the black hole information paradox. We show that in these models, the physical resources required to send a superluminal signal scale polynomially with the resources needed to speed up Grover's algorithm. Hence the no-signaling principle is equivalent to the inability to solve NP-hard problems efficiently by brute force within the classes of theories analyzed.

quant-ph↗

arXiv2016

The Computational Complexity of Ball Permutations

Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.

quant-ph↗

arXiv2016

On the complexity of probabilistic trials for hidden satisfiability problems

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.

cs.CC↗

arXiv2016

Complexity classification of two-qubit commuting hamiltonians

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

quant-ph↗

arXiv2014

The space "just above" BQP

We explore the space "just above" BQP by defining a complexity class PDQP (Product Dynamical Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform measurements that do not collapse the wavefunction. This (non-physical) model of computation can efficiently solve problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. Furthermore, it can search an unstructured N-element list in $\tilde O$(N^{1/3}) time, but no faster than Ω(N^{1/4}), and hence cannot solve NP-hard problems in a black box manner. In short, this model of computation is more powerful than standard quantum computation, but only slightly so. Our work is inspired by previous work of Aaronson on the power of sampling the histories of hidden variables. However Aaronson's work contains an error in its proof of the lower bound for search, and hence it is unclear whether or not his model allows for search in logarithmic time. Our work can be viewed as a conceptual simplification of Aaronson's approach, with a provable polynomial lower bound for search.

quant-ph↗

arXiv2014

Generation of Universal Linear Optics by Any Beamsplitter

In 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on the single-photon subspace with m>=3 modes. (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics. Theoretically, it means that one cannot produce "intermediate" models of linear optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.

quant-ph↗

arXiv2013

Psi-Epistemic Theories: The Role of Symmetry

Formalizing an old desire of Einstein, "psi-epistemic theories" try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states." Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out, by proving no-go theorems analogous to the Bell Inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant psi-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic psi-epistemic theory must be "measurement contextual" in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under tensor product." In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that satisfy two reasonable conditions: (1) symmetry under unitary transformations, and (2) "maximum nontriviality" (meaning that the probability distributions corresponding to any two non-orthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally-nontrivial psi-epistemic theories in every finite dimension d.

quant-ph↗

arXiv2011

Caching and Interpolated Likelihoods: Accelerating Cosmological Monte Carlo Markov Chains

We describe a novel approach to accelerating Monte Carlo Markov Chains. Our focus is cosmological parameter estimation, but the algorithm is applicable to any problem for which the likelihood surface is a smooth function of the free parameters and computationally expensive to evaluate. We generate a high-order interpolating polynomial for the log-likelihood using the first points gathered by the Markov chains as a training set. This polynomial then accurately computes the majority of the likelihoods needed in the latter parts of the chains. We implement a simple version of this algorithm as a patch (InterpMC) to CosmoMC and show that it accelerates parameter estimatation by a factor of between two and four for well-converged chains. The current code is primarily intended as a "proof of concept", and we argue that there is considerable room for further performance gains. Unlike other approaches to accelerating parameter fits, we make no use of precomputed training sets or special choices of variables, and InterpMC is almost entirely transparent to the user.

astro-ph.CO↗