Adam Bouland

ψ-epistemic theories: The role of symmetry

Formalizing an old desire of Einstein, “ψ-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 ψ-epistemic theory for Hilbert space dimension d = 2, but also showed that any deterministic ψ-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 ψ-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 ψ-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 ψ-epistemic theories in every finite dimension d.

On tractable parameterizations of graph isomorphism

The fixed-parameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as tree-width, genus and maximum degree. We show that graph isomorphism is fixed-parameter tractable when parameterized by the tree-depth of the graph. We also extend this result to a parameter generalizing both tree-depth and max-leaf-number by deploying new variants of cops-and-robbers games.

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.

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 Grovers 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.

The computational complexity of ball permutations

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 use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. 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, and that a nondeterministic version of this model is NP-complete.

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. 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 which separate SZK from PZK, NISZK from NIPZK and PZK from 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.

Rescuing complementarity with little drama

A bstractThe 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.

Establishing quantum advantage

What are quantum computers good for? This essay reviews the progress toward proving a quantum advantage over classical computing.

On the complexity and verification of quantum 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.