The Power of Unentanglement

The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. * We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. * We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one

Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\mathsf{SAT}$ problem, the state-of-the-art is that the problem cannot be solved by random-access machines in $n^c$ time and $n^{o(1)}$ space simultaneously for $c < 2\cos(\frac{\pi}{7}) \approx 1.801$. We extend this lower bound approach to the quantum and randomized domains. Combining Grover's algorithm with components from $\mathsf{SAT}$ time-space lower bounds, we show that there are problems verifiable in $O(n)$ time with quantum Merlin-Arthur protocols that cannot be solved in $n^c$ time and $n^{o(1)}$ space simultaneously for $c < \frac{3+\sqrt{3}}{2} \approx 2.366$, a super-quadratic time lower bound. This result and the prior work on $\mathsf{SAT}$ can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in $O(n)$ time with (classical) Merlin-Arthur protocols that cannot be solved in $n^c$ randomized time and $n^{o(1)}$ space simultaneously for $c < 1.465$, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to $c < 1.5$.Comment: 38 pages, 5 figures. To appear in ITCS 202

Thermodynamic work from operational principles

In recent years we have witnessed a concentrated effort to make sense of thermodynamics for small-scale systems. One of the main difficulties is to capture a suitable notion of work that models realistically the purpose of quantum machines, in an analogous way to the role played, for macroscopic machines, by the energy stored in the idealisation of a lifted weight. Despite of several attempts to resolve this issue by putting forward specific models, these are far from capturing realistically the transitions that a quantum machine is expected to perform. In this work, we adopt a novel strategy by considering arbitrary kinds of systems that one can attach to a quantum thermal machine and seeking for work quantifiers. These are functions that measure the value of a transition and generalise the concept of work beyond the model of a lifted weight. We do so by imposing simple operational axioms that any reasonable work quantifier must fulfil and by deriving from them stringent mathematical condition with a clear physical interpretation. Our approach allows us to derive much of the structure of the theory of thermodynamics without taking as a primitive the definition of work. We can derive, for any work quantifier, a quantitative second law in the sense of bounding the work that can be performed using some non-equilibrium resource by the work that is needed to create it. We also discuss in detail the role of reversibility and correlations in connection with the second law. Furthermore, we recover the usual identification of work with energy in degrees of freedom with vanishing entropy as a particular case of our formalism. Our mathematical results can be formulated abstractly and are general enough to carry over to other resource theories than quantum thermodynamics.Comment: 22 pages, 4 figures, axioms significantly simplified, more comprehensive discussion of relationship to previous approache

Testing product states, quantum Merlin-Arthur games and tensor optimisation

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC

Approximation, Proof Systems, and Correlations in a Quantum World

This thesis studies three topics in quantum computation and information: The approximability of quantum problems, quantum proof systems, and non-classical correlations in quantum systems. In the first area, we demonstrate a polynomial-time (classical) approximation algorithm for dense instances of the canonical QMA-complete quantum constraint satisfaction problem, the local Hamiltonian problem. In the opposite direction, we next introduce a quantum generalization of the polynomial-time hierarchy, and define problems which we prove are not only complete for the second level of this hierarchy, but are in fact hard to approximate. In the second area, we study variants of the interesting and stubbornly open question of whether a quantum proof system with multiple unentangled quantum provers is equal in expressive power to a proof system with a single quantum prover. Our results concern classes such as BellQMA(poly), and include a novel proof of perfect parallel repetition for SepQMA(m) based on cone programming duality. In the third area, we study non-classical quantum correlations beyond entanglement, often dubbed "non-classicality". Among our results are two novel schemes for quantifying non-classicality: The first proposes the new paradigm of exploiting local unitary operations to study non-classical correlations, and the second introduces a protocol through which non-classical correlations in a starting system can be "activated" into distillable entanglement with an ancilla system. An introduction to all required linear algebra and quantum mechanics is included.Comment: PhD Thesis, 240 page

Communication Complexity of Set-Disjointness for All Probabilities

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness

Generalized Quantum Arthur-Merlin Games

This paper investigates the role of interaction and coins in public-coin quantum interactive proof systems (also called quantum Arthur-Merlin games). While prior works focused on classical public coins even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the public coins can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. First, it is proved that the class of two-turn quantum Arthur-Merlin games with quantum public coins, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interactions prior to the communications of the qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper provides a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking the properties related to quantum circuits, and is of independent interest. It is further proved that the class qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems with perfect completeness gives new bounds for standard well-studied classes of two-turn interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of interaction and public coins in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the class of problems having an m-turn quantum Arthur-Merlin proof system is either equal to PSPACE or equal to the class of problems having a two-turn quantum Arthur-Merlin game of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.Comment: 31 pages + cover page, the proof of Lemma 27 (Lemma 24 in v1) is corrected, and a new completeness result is adde

Space-Time Circuit-to-Hamiltonian Construction and Its Applications

The circuit-to-Hamiltonian construction translates dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman-Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each interacting qubit. This makes it possible to capture the spatio-temporal structure of the original quantum circuit into features of the circuit Hamiltonian. The construction is inspired by the original two-dimensional interacting fermionic model (see http://link.aps.org/doi/10.1103/PhysRevA.63.040302) We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lower-bounded, partially using results on mixing times of Markov chains, so that the applications of this construction for QMA (and partially for quantum adiabatic computation) go through. For one-dimensional quantum circuits, the dynamics generated by the circuit Hamiltonian corresponds to diffusion of a string around the torus.Comment: 27 pages, 5 figure