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Closed Timelike Curves Make Quantum and Classical Computing Equivalent
"... While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: b ..."
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While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class PSPACE, consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves an open problem proposed by one of us in 2005, and gives an essentially complete understanding of computational complexity in the presence of CTCs. Following the work of Deutsch, we treat a CTC as simply a region of spacetime where a “causal consistency ” condition is imposed, meaning that Nature has to produce a (probabilistic or quantum) fixedpoint of some evolution operator. Our conclusion is then a consequence of the following theorem: given any quantum circuit (not necessarily unitary), a fixedpoint of the circuit can be (implicitly) computed in polynomial space. This theorem might have independent applications in quantum information. 1
Complexity classification of local Hamiltonian problems
, 2013
"... The calculation of groundstate energies of physical systems can be formalised as the klocal Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question ..."
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The calculation of groundstate energies of physical systems can be formalised as the klocal Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensedmatter physics. In this work we characterise the complexity of this problem for all 2local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P; NPcomplete; polynomialtime equivalent to the Ising model with transverse magnetic fields; or QMAcomplete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1local terms; for example, we prove for the first time QMAcompleteness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2local interactions: for any constant k, all klocal qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P; polynomialtime equivalent to the Ising model with transverse magnetic fields; or QMAcomplete. These results are a quantum analogue of Schaefer’s dichotomy theorem for boolean constraint satisfaction problems. 1
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
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One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing
Stronger methods of making quantum interactive proofs perfectly complete
 In ITCS ’13, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science
, 2013
"... ar ..."
Multiprover quantum MerlinArthur proof systems with small gap
, 2012
"... This paper studies multipleproof quantum MerlinArthur (QMA) proof systems in the setting when the completenesssoundness gap is small. Small means that we only lowerbound the gap with an inverseexponential function of the input length, or with an even smaller function. Using the protocol of Blie ..."
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Cited by 4 (2 self)
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This paper studies multipleproof quantum MerlinArthur (QMA) proof systems in the setting when the completenesssoundness gap is small. Small means that we only lowerbound the gap with an inverseexponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [BT09], we show that in this case the proof system has the same expressive power as nondeterministic exponential time (NEXP). Since singleproof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between single and multiprover proof systems in the ‘smallgap setting’, under the assumption that EXP 6=NEXP. This implies, among others, the nonexistence of certain operators called disentanglers (defined by Aaronson et al. [ABD+09]), with good approximation parameters. We also show that in this setting the proof system has the same expressive power if we restrict the verifier to be able to perform only Bellmeasurements, i.e., using a BellQMA verifier. This is not known to hold in the usual setting, when the gap is bounded by an inversepolynomial function of the input length. To show this we use the protocol of Chen and Drucker [CD10]. The only caveat here is that we need at least a linear amount of proofs to achieve the power of NEXP, while in the previous setting two proofs were enough. We also study the case when the prooflengths are only logarithmic in the input length and observe that in some cases the expressive power decreases. However, we show that it doesn’t decrease further if we make the proof lengths to be even shorter. 1
Computational Distinguishability of Quantum Channels
, 2009
"... The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that h ..."
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The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixedunitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixedunitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.