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ACM Symposium on Theory of Computing (STOC)

Quantum lower bounds by quantum arguments

Andris Ambainis

Published May 1, 2000• 122 citations
Publisher

Abstract

We propose a new method for proving lower bounds on quantum query algorithms.Instead of a classical adversary that runs the algorithm with one input and then modifies the input we use a quantum adversary that runs the input with a superposition of inputs.Using this method, we prove two new ~2(x/-N) lower bounds on AND of ORs and inverting a permutation and also provide more uniform proofs for some known lower bounds which have been previously proven via variety of different techniques.

DOI
10.1145/335305.335394

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We propose a new method for proving lower bounds on quantum query algorithms.Instead of a classical adversary that runs the algorithm with one input and then modifies the input we use a quantum adversary that runs the input with a superposition of inputs.Using this method, we prove two new ~2(x/-N) lower bounds on AND of ORs and inverting a permutation and also provide more uniform proofs for some known lower bounds which have been previously proven via variety of different techniques.
We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω(N)Ω(\sqrt{N})Ω(N​) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via variety of different techniques.
The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations interms of weight schemes, eigen values, and Kolmogorov complexity. All these formulations rely on the principlethat if an algorithm successfully computes a function then, in particular, itis able to distinguish between inputs which map to different values.
We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in O(min⁡(nC0(f),nC1(f)))O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))O(min(nC0​(f)​,nC1​(f)​)), where C0,C1C_0, C_1C0​,C1​ is the certificate complexity and n is the size of the input.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.
The quantum adversary method is a versatile method for proving lower bounds on quantum algorithms. It yields tight bounds for many computational problems, is robust in having many equivalent formulations, and has natural connections to classical lower bounds. A further nice property of the adversary method is that it behaves very well with respect to composition of functions. We generalize the adversary method to include costs--each bit of the input can be given an arbitrary positive cost representing the difficulty of querying that bit. We use this generalization to exactly capture the adversary bound of a composite function in terms of the adversary bounds of its component functions. Our results generalize and unify previously known composition properties of adversary methods, and yield as a simple corollary the Omega(sqrt{n}) bound of Barnum and Saks on the quantum query complexity of read-once functions.