38 research outputs found
Trading Inverses for an Irrep in the Solovay-Kitaev Theorem
No full textThe 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 epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], 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 [Dawson and Nielsen, 2006; Kuperberg, 2015]
Generation of universal linear optics by any beam splitter
No full textIn 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beam splitters and phase shifters. Here we show that any single beam splitter 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 phase shifter.) Experimentally, this means that one does not need tunable beam splitters or phase shifters for universality: any nontrivial beam splitter 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 beam splitters and phase shifters: 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.National Science Foundation (U.S.) (Grant 0844626)National Science Foundation (U.S.) (Alan T. Waterman Award)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374)National Science Foundation (U.S.). Center for Science of Information (Grant Agreement CCF-0939370
Complexity Classification of Two-Qubit Commuting Hamiltonians
No full textWe 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
Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract)
No full textThe AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity.
Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be "scrambling" - i.e. the expectation value of local operators equilibrates in polynomial time - whereas the gravity theory is not, because the interiors of certain black holes known as "wormholes" do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. So what could be the CFT dual to wormhole volume? Susskind’s proposed resolution was to equate the wormhole volume with the quantum circuit complexity of the CFT state. From a computer science perspective, circuit complexity seems like an unusual choice because it should be difficult to compute, in contrast to physical quantities such as wormhole volume.
We show how to create pseudorandom quantum states in the CFT, thereby arguing that their quantum circuit complexity is not "feelable", in the sense that it cannot be approximated by any efficient experiment. This requires a specialized construction inspired by symmetric block ciphers such as DES and AES, since unfortunately existing constructions based on quantum-resistant one way functions cannot be used in the context of the wormhole growth paradox as only very restricted operations are allowed in the CFT. By contrast we argue that the wormhole volume is "feelable" in some general but non-physical sense. The duality between a "feelable" quantity and an "unfeelable" quantity implies that some aspect of this duality must have exponential complexity. More precisely, it implies that either the dictionary is exponentially complex, or else the quantum gravity theory is exponentially difficult to simulate on a quantum computer.
While at first sight this might seem to justify the discomfort of complexity theorists with equating computational complexity with a physical quantity, a further examination of our arguments shows that any resolution of the wormhole growth paradox must equate wormhole volume to an "unfeelable" quantity, leading to the same conclusions. In other words this discomfort is an inevitable consequence of the paradox
"Quantum Supremacy" and the Complexity of Random Circuit Sampling
No full textA critical goal for the field of quantum computation is quantum supremacy - a demonstration of any quantum computation that is prohibitively hard for classical computers. It is both a necessary milestone on the path to useful quantum computers as well as a test of quantum theory in the realm of high complexity. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, called Random Circuit Sampling (RCS).
While RCS was defined with experimental realization in mind, we give strong complexity-theoretic evidence for the classical hardness of RCS, placing it on par with the best 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 also satisfies an anti-concentration property, namely that errors in estimating output probabilities are small with respect to the probabilities themselves. This makes RCS the first proposal for quantum supremacy with both of these properties. We also give a natural condition under which an existing statistical measure, cross-entropy, verifies RCS, as well as describe a new verification measure which in some formal sense maximizes the information gained from experimental samples
Complexity Classification of Conjugated Clifford Circuits
No full textClifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/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
ψ-epistemic theories: The role of symmetry
No full textFormalizing 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 nonorthogonal 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.National Science Foundation (U.S.) (Grant 0844626)National Science Foundation (U.S.) (STC Grant)Alfred P. Sloan Foundation (Fellowship)Alan T. Waterman AwardNational Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374)National Science Foundation (U.S.) (Center for Science of Information Grant Agreement CCF-0939370)MIT Summer Research Progra
Grover Search and the No-Signaling Principle
No full textTwo 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.National Science Foundation (U.S.) (Alan T. Waterman Award Grant 1249349)DuBridge Postdoctoral FellowshipNational Science Foundation (U.S.) (Physics Frontiers Center, Institute for Quantum Information and Matter. Grant PHY-1125565)Gordon and Betty Moore Foundation (Grant GBMF-12500028)United States. Dept. of Energy. Office of High Energy Physics (Award DE-SC0011632)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374
On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems
No full textWhat 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
Quantum Pseudoentanglement
No full textEntanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement", a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states - first defined by Ji, Liu and Song - which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to log n across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously
