Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Constructing Steane code fault-tolerant gates

Austin G. Fowler, Simon J. Devitt, Lloyd C. L. Hollenberg

Published Feb 23, 2005• 0 citations

Abstract

We construct fault-tolerant approximations of rotation gates required by Shor's algorithm using only fault-tolerant gates that can be applied to the 7-qubit Steane code. A general scaling law of how rapidly these fault-tolerant approximations converge to arbitrary single-qubit gates is also determined.

DOI: 10.1117/12.583421

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Constructing Steane code fault-tolerant gates

Austin G. Fowler, Simon J. Devitt, Lloyd C. L. Hollenberg

Published Feb 23, 2005• 0 citations

Publisher

Abstract

DOI: 10.1117/12.583421

Related papers

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Scalability of Shor’s algorithm with a limited set of rotation gates

Austin G. Fowler, Lloyd C. L. Hollenberg

Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $\ensuremath{\pi}∕{2}^{2L}$ where $L$ is the binary length of the integer $N$ to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $\ensuremath{\pi}∕{2}^{d}$ rotation gate to within $\ensuremath{\delta}=O(1∕{2}^{d})$ currently requires both a number of qubits and a number of fault-tolerant gates that grows polynomially with $d$ . In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large integers. Consequently, we study in detail the effect of using only controlled rotation gates with $d$ less than or equal to some ${d}_{\mathrm{max}}$ . It is found that integers up to length ${L}_{\mathrm{max}}=O({4}^{{d}_{\mathrm{max}}})$ can be factored without significant performance penalty implying that the cumbersome techniques of fault-tolerant computation only need to be used to create controlled rotation gates of magnitude $\ensuremath{\pi}∕64$ if integers thousands of bits long are desired factored. Explicit fault-tolerant constructions of such gates are also discussed.Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $\ensuremath{\pi}∕{2}^{2L}$ where $L$ is the binary length of the integer $N$ to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $\ensuremath{\pi}∕{2}^{d}$ rotation gate to within $\ensuremath{\delta}=O(1∕{2}^{d})$ currently requires both a number of qubits and a number of fault-tolerant gates that grows polynomially with $d$. In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large integers. Consequently, we study in detail the effect of using only controlled rotation gates with $d$ less than or equal to some ${d}_{\mathrm{max}}$. It is found that integers up to length ${L}_{\mathrm{max}}=O({4}^{{d}_{\mathrm{max}}})$ can be factored without significant performance penalty implying that the cumbersome techniques of fault-tolerant computation only need to be used to create controlled rotation gates of magnitude $\ensuremath{\pi}∕64$ if integers thousands of bits long are desired factored. Explicit fault-tolerant constructions of such gates are also discussed.

arXiv:quant-ph/0306018Physical Review ASep 29, 200462 citations

Fault-tolerant logical operations for qubits encoded by Calderbank-Shor-Steane codes are discussed, with emphasis on methods that apply to codes of high rate, encoding

k

qubits per block with

k>1

. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into

X

and

Z

parts. Optimized methods to move logical qubits around and to achieve controlled-NOT and Toffoli gates are discussed. It is found that the number of time steps required to complete a fault-tolerant quantum computation is the same when

k>1

as when

k=1

Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $\ensuremath{\pi}∕{2}^{2L}$ where

L

is the binary length of the integer

N

to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $\ensuremath{\pi}∕{2}^{d}$ rotation gate to within $\ensuremath{\delta}=O(1∕{2}^{d})$ currently requires both a number of qubits and a number of fault-tolerant gates that grows polynomially with

d

. In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large integers. Consequently, we study in detail the effect of using only controlled rotation gates with

d

less than or equal to some

{d}_{\mathrm{max}}

. It is found that integers up to length

{L}_{\mathrm{max}}=O({4}^{{d}_{\mathrm{max}}})

can be factored without significant performance penalty implying that the cumbersome techniques of fault-tolerant computation only need to be used to create controlled rotation gates of magnitude $\ensuremath{\pi}∕64$ if integers thousands of bits long are desired factored. Explicit fault-tolerant constructions of such gates are also discussed.