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arXiv:quant-ph/0703237 · IEEE Conference on Computational Complexity (CCC)

The Multiplicative Quantum Adversary

Robert Špalek

Published Jun 1, 2008• 29 citations
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Abstract

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.

DOI
10.1109/ccc.2008.9

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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.
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.
We study the two-weight decision version of quantum approximate counting: given oracle access to x∈{0,1}Nx\in\{0,1\}^Nx∈{0,1}N, distinguish ∣x∣=M|x|=M∣x∣=M from ∣x∣=M+Δ|x|=M+Δ∣x∣=M+Δ with success probability 1/2+ζ1/2+ζ1/2+ζ. Using the multiplicative adversary method, we prove Ω(max⁡{ζ(N−M)(M+Δ)/Δ,ζN/Δ})Ω\left(\max\left\{ζ\sqrt{(N-M)(M+Δ)}/Δ,\sqrt{ζN/Δ}\right\}\right)Ω(max{ζ(N−M)(M+Δ)​/Δ,ζN/Δ​}). The same parameter dependence follows from the polynomial-method characterization of the two-layer symmetric function by Podder, Yao, and Ye. Our contribution is a multiplicative-adversary derivation that tracks the progress produced by individual oracle queries. For the first term, after complementing the input if necessary, we assume M+Δ≤N−MM+Δ\le N-MM+Δ≤N−M. We use the Hamming-layer subspaces from the eigenspace method of Ambainis, Spalek, and de Wolf and compose their adjacent-layer unitary maps to relate the two nonadjacent promise layers. After fixing the queried coordinate, the analysis block-diagonalizes into four-dimensional subspaces. An exact calculation of the one-query progress ratio gives the first lower bound. The same estimate also implies ∥(I−Π^bad)∣ΨT⟩∥2=O(T2Δ2/((N−M)(M+Δ)))\left\|(I-\widehatΠ_{\mathrm{bad}})\lvertΨ^T\rangle\right\|^2=O\left(T^2Δ^2/((N-M)(M+Δ))\right)​(I−Πbad​)∣ΨT⟩​2=O(T2Δ2/((N−M)(M+Δ))) for the coherent input superposition used in the adversary argument. For the second term, we prove directly using a three-eigenvalue multiplicative adversary that unique OR on nnn bits with success probability 1/2+ζ1/2+ζ1/2+ζ requires Ω(ζn)Ω(\sqrt{ζn})Ω(ζn​) queries, and then reduce unique OR to the two-weight counting problem.
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 show that quantum query complexity satisfies a strong direct product theorem. This means that computing kkk copies of a function with less than kkk times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in kkk. For a boolean function fff we also show an XOR lemma -- computing the parity of kkk copies of fff with less than kkk times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.
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.