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The complexity class PDQP SPUR Final Paper: Summer 2013

Mitchell Lee, Adam Bouland, Scott Aaronson

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Quantum computers are believed to be strictly more computationally powerful than classical computers, but not so much more powerful that they can solve NP-complete problems eciently. This is because of the result that a quantum computer takes ( N 1=2 ) time to search an unstructured N-element list for a particular marked item, as opposed to poly(logN) time for a nondeterministic computer. On the other hand, many seemingly innocuous modications of quantum mechanics increase the power of quantum computers drastically enough that they can solve NP-complete problems eciently [3]. This paper denes a model of computation slightly more powerful than quantum computation, but only slightly so. In particular, we show that by allowing \non-collapsing measurements, we can solve eciently problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. We can also search an unstructured N-element list for a particular marked item in ~ O(N 1=3 ) time, but no faster than ( N 1=4 ).

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Quantum computers are believed to be strictly more computationally powerful than classical computers, but not so much more powerful that they can solve NP-complete problems eciently. This is because of the result that a quantum computer takes ( N 1=2 ) time to search an unstructured N-element list for a particular marked item, as opposed to poly(logN) time for a nondeterministic computer. On the other hand, many seemingly innocuous modications of quantum mechanics increase the power of quantum computers drastically enough that they can solve NP-complete problems eciently [3]. This paper denes a model of computation slightly more powerful than quantum computation, but only slightly so. In particular, we show that by allowing \non-collapsing measurements, we can solve eciently problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. We can also search an unstructured N-element list for a particular marked item in ~ O(N 1=3 ) time, but no faster than ( N 1=4 ).
We explore the space "just above" BQP by defining a complexity class naCQP (non-adaptive Collapse-free 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 (non-adaptive) 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 Õ(N1/3) time, but no faster than Ω(N1/4), 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. This is surprising as most modifications of BQP increase the power of quantum computation to NP or beyond.
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 O~\tilde OO~(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.
Aaronson, Bouland, Fitzsimons and Lee introduced the complexity class PDQP (which was original labeled naCQP), an alteration of BQP enhanced with the ability to obtain non-collapsing measurements, samples of quantum states without collapsing them. Although PDQP contains SZK, it still requires Ω(N1/4)Ω(N^{1/4})Ω(N1/4) queries to solve unstructured search. We formulate an alternative equivalent definition of PDQP, which we use to prove the positive weighted adversary lower-bounding method, establishing multiple tighter bounds and a trade-off between queries and non-collapsing measurements. We utilize the technique in order to analyze the query complexity of the well-studied majority and element distinctness problems. Additionally, we prove a tight Θ(N1/3)Θ(N^{1/3})Θ(N1/3) bound on search. Furthermore, we use the lower-bound to explore PDQP under query restrictions, finding that when combined with non-adaptive queries, we limit the speed-up in several cases.
This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory satisfying a reasonable axiom called "indifference to the identity," we could solve the Graph Isomorphism and Approximate Shortest Vector problems in polynomial time, as well as an oracle problem that is known to require quantum exponential time. We could also search an N-item database using O(N^{1/3}) queries, as opposed to O(N^{1/2}) queries with Grover's search algorithm. On the other hand, the N^{1/3} bound is optimal, meaning that we could probably not solve NP-complete problems in polynomial time. We thus obtain the first good example of a model of computation that appears slightly more powerful than the quantum computing model.
This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.
The central problem in computer science is the conjecture that two complexity classes, P (polynomial time) and NP (nondeterministic polynomial time-roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: P not equal NP. As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all #P problems, a computationally intractable class, in polynomial time. Specifically, Witten [Witten, E. (1989) Commun. Math. Phys. 121, 351-391] has identified expectation values in a certain SU(2)-field theory with values of the Jones polynomial [Jones, V. (1985) Bull. Am. Math. Soc. 12, 103-111] that are #P-hard [Jaeger, F., Vertigen, D. & Welsh, D. (1990) Math. Proc. Comb. Philos. Soc. 108, 35-53]. This suggests that some physical system whose effective Lagrangian contains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving NP or even #P-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.