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arXiv:quant-ph/9808027

Parallel Quantum Computation and Quantum Codes

Cristopher Moore, Martin Nilsson

Published Aug 17, 1998• 16 citations
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Abstract

We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum error-correcting codes, or more generally any circuit consisting of controlled-not gates, controlled pi-shifts, and Hadamard gates. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we conjecture that an even simpler `staircase' circuit cannot be parallelized to less than linear depth, and might be used to prove that QNC < QP.

DOI
10.48550/arxiv.quant-ph/9808027

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We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum error-correcting codes, or more generally any circuit consisting of controlled-not gates, controlled pi-shifts, and Hadamard gates. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we conjecture that an even simpler `staircase' circuit cannot be parallelized to less than linear depth, and might be used to prove that QNC < QP.
We study the class QNC of efficient parallel quantum circuits, the quantum analog of NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum error-correcting codes, or, more generally, any circuit consisting of controlled-not gates, controlled π\piπ-shifts, and Hadamard gates. Finally, while we note the exact quantum Fourier transform can be parallelized to linear depth, we conjecture that neither itnor a simpler "staircase" circuit can be parallelized to less than this.
Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting transversal gates. However, a natural class of quantum codes, which has been well-studied classically, has not yet been treated: those which can be quickly encoded and decoded. This problem concerns the channel capacity setting, where a noise channel sits between perfect encoding and unencoding/decoding operations; this is the setting that is relevant for communication between fault-tolerant quantum computers. In this work, we construct asymptotically good quantum codes that can be encoded and unencoded by quantum circuits of logarithmic depth and consisting of a linear total number of gates. The classical decoding algorithms also run in logarithmic depth and use O(nlog⁡n)\mathcal{O}(n \log n)O(nlogn) gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.
We exhibit some simple gadgets useful in designing shallow parallel circuits for quantum algorithms. We prove that any quantum circuit composed entirely of controlled-not gates or of diagonal gates can be parallelized to logarithmic depth, while circuits composed of both cannot. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we exhibit a simple quantum circuit related to it that we believe cannot be parallelized to less than linear depth, and therefore might be used to prove that QNC < QP.
We exhibit a quantum circuit that performs the Quantum Fourier Transform on nnn qubits in $ (n)$ depth. Thus, a parallel quantum computer can carry out the QFT in linear time. We conjecture that this can, in fact, be reduced to $ ({\rm log} n)$ depth, which would place the QFT in the class {\bf QNC} of problems solvable in logarithmic parallel time.
We study the quantum complexity class QNC^0_f of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates (called QNC^0_f circuits). Our main result is that the quantum OR operation is in QNC^0_f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC^0 \subsetneq AC^0 \subsetneq TC^0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC^0_f = QAC^0_f = QTC^0_f. Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This implies the size difference between the QNC^0_f and QTC^0_f circuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC^0_f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC^0_f oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC^0_f circuit with gates for the quantum Fourier transform.
The main focus of this work is to implement a program that can be used to parallelise quantum circuits. This is done by translating the quantum circuits to the Measurement Based Quantum Computing (MBQC) model, applying a number of optimising techniques to the computation in MBQC, and by translating it back to a quantum circuit. As a side result of implementing this method, a new algorithm for creating and optimising an MBQC graph from a quantum circuit is designed. The implemented program is used to investigate quantum circuits known from the literature: the QFT, and a specific type of circuits. We show that our program will parallelise the circuit to logarithmic depth, and increase the depth of QFT by a logarithmic factor. Finally, three new classes of quantum circuits are presented: the Toffoli staircase circuit, the Toffoli + CNOT staircase circuit, and a circuits consisting of quantum gates ∧Z, CNOT , ω , Z(α), and J(−2 ). In the computational experiments we perform, those classes of circuits will benefit from the parallelisation method implemented in this dissertation.