Structured Error Recovery for Codeword-Stabilized Quantum Codes

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Abstract

Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical non-linear codes. For an n-qubit quantum code correcting errors on up to t qubits, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. In this paper, we suggest an error grouping technique that allows to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3^t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.

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Mixed State Entanglement and Quantum Error Correction

William Wootters, John Smolin, David P. DiVincenzo … (1996)

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state \(|\xi\rangle\) can be transmitted at some rate Q through a noisy channel \(\chi\) without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state \(\hat{M}(\chi)\) (obtained by sharing halves of EPR pairs through a channel \(\chi\)) yields a QECC on \(\chi\) with rate \(Q=D\), and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts \(D_1(M)\) and \(D_2(M)\) that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for \(E(M)\) when \(M\) is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic \(Q\) (or \(D\)) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.

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Scheme for reducing decoherence in quantum computer memory.

Shor (1995)

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Elementary gates for quantum computation

(2010)

We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values \((x,y)\) to \((x,x \oplus y)\)) is universal in the sense that all unitary operations on arbitrarily many bits \(n\) (U(\(2^n\))) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for \(n\)-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary \(n\)-bit unitary operations.