The Equivalence of Sampling and Searching

In a sampling problem, we are given an input x, and asked to sample approximately from a probability distribution D_x. In a search problem, we are given an input x, and asked to find a member of a nonempty set A_x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity and algorithmic information theory to show that sampling and search problems are essentially equivalent. More precisely, for any sampling problem S, there exists a search problem R_S such that, if C is any "reasonable" complexity class, then R_S is in the search version of C if and only if S is in the sampling version. As one application, we show that SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve. A second application is that, assuming a plausible conjecture, there exists a search problem R that can be solved using a simple linear-optics experiment, but that cannot be solved efficiently by a classical computer unless the polynomial hierarchy collapses. That application will be described in a forthcoming paper with Alex Arkhipov on the computational complexity of linear optics.Comment: 16 page

Towards practical classical processing for the surface code: timing analysis

Topological quantum error correction codes have high thresholds and are well suited to physical implementation. The minimum weight perfect matching algorithm can be used to efficiently handle errors in such codes. We perform a timing analysis of our current implementation of the minimum weight perfect matching algorithm. Our implementation performs the classical processing associated with an nxn lattice of qubits realizing a square surface code storing a single logical qubit of information in a fault-tolerant manner. We empirically demonstrate that our implementation requires only O(n^2) average time per round of error correction for code distances ranging from 4 to 512 and a range of depolarizing error rates. We also describe tests we have performed to verify that it always obtains a true minimum weight perfect matching.Comment: 13 pages, 13 figures, version accepted for publicatio

Degenerate Quantum Codes for Pauli Channels

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.Comment: Introduction changed to give more motivation and background. Figure 1 replace

Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer

A scheme for linear optical implementation of fault-tolerant quantum computation is proposed, which is based on an error-detecting code. Each computational step is mediated by transfer of quantum information into an ancilla system embedding error-detection capability. Photons are assumed to be subjected to both photon loss and depolarization, and the threshold region of their strengths for scalable quantum computation is obtained, together with the amount of physical resources consumed. Compared to currently known results, the present scheme reduces the resource requirement, while yielding a comparable threshold region.Comment: 9 pages, 7 figure

Fault-Tolerant Error Correction with Efficient Quantum Codes

We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.Comment: 12 pages REVTeX, 1 ps figure included. Minor additions and revision

Restrictions on Transversal Encoded Quantum Gate Sets

Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.Comment: 4 pages, v2: minor change

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently

Oblivious remote state preparation

We consider remote state preparation protocols for a set of pure states whose projectors form a basis for operators acting on the input Hilbert space. If a protocol (1) uses only forward communication and entanglement, (2) deterministically prepares an exact copy of the state, and (3) does so obliviously -- without leaking further information about the state to the receiver -- then the protocol can be modified to require from the sender only a single specimen of the state. Furthermore, the original protocol and the modified protocol use the same amount of classical communication. Thus, under the three conditions stated, remote state preparation requires at least as much classical communication as teleportation, as Lo has conjectured [PRA 62 (2000) 012313], which is twice the expected classical communication cost of some existing nonoblivious protocols