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arXiv:1012.2112 · IEEE Conference on Computational Complexity (CCC)

Symmetry-Assisted Adversaries for Quantum State Generation

Andris Ambainis, Loïck Magnin, Martin Roetteler, Jérémie Roland

Published Jun 1, 2011• 50 citations
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Abstract

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the GRAPH-ISOMORPHISM problem. We show that for the related problem of INDEX-ERASURE our method leads to a lower bound of square root of N which matches an upper bound obtained via reduction to quantum search on N elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for INDEX-ERASURE by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to Spalek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by Spalek to quantum state generation problems.

DOI
10.1109/ccc.2011.24

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We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the GRAPH-ISOMORPHISM problem. We show that for the related problem of INDEX-ERASURE our method leads to a lower bound of square root of N which matches an upper bound obtained via reduction to quantum search on N elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for INDEX-ERASURE by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to Spalek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by Spalek to quantum state generation problems.
We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.
The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independent of the function. This new result therefore provides insight on the reason why in some cases the adversary method is stronger than the polynomial method. It also reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplicative adversary method.
The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight Ω(√n) lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the so-called automorphism principle (Høyer et al. STOC 2007) to the general adversary method for state conversion problems (Lee et al. STOC 2011), which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the well-known Krein parameters of association schemes. Krein parameters are usually hard to determine, nevertheless, we give a novel way of computing certain Krein parameters of a commutative association scheme defined over partial permutations. We believe the study of this association scheme may also be of independent interest.
Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a relational formulation of the multiplicative adversary method by Jeffery and Zur which we prove, via an equivalent formulation, satisfies a strong selective direct product property and captures any query lower bound for functions proven by negative-weights adversaries. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean-valued functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.
We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.