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arXiv:1210.1782 · Journal of Experimental and Theoretical Physics Letters

Revisiting the hopes for scalable quantum computation

M. I. Dyakonov

Published Dec 1, 2013• 9 citations
arXivHTMLPDF

Abstract

The hopes for scalable quantum computing rely on the “threshold theorem”: once the error per qubit per gate is below a certain value, the methods of quantum error correction allow indefinitely long quantum computations. The proof is based on a number of assumptions, which are supposed to be satisfied exactly , like axioms, e.g., zero undesired interactions between qubits, etc. However, in the physical world no continuous quantity can be exactly zero, it can only be more or less small. Thus the “error per qubit per gate” threshold must be complemented by the required precision with which each assumption should be fulfilled. In the absence of this crucial information, the prospects of scalable quantum computing remain uncertain.

DOI
10.1134/s0021364013210042

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The hopes for scalable quantum computing rely on the “threshold theorem”: once the error per qubit per gate is below a certain value, the methods of quantum error correction allow indefinitely long quantum computations. The proof is based on a number of assumptions, which are supposed to be satisfied exactly , like axioms, e.g., zero undesired interactions between qubits, etc. However, in the physical world no continuous quantity can be exactly zero, it can only be more or less small. Thus the “error per qubit per gate” threshold must be complemented by the required precision with which each assumption should be fulfilled. In the absence of this crucial information, the prospects of scalable quantum computing remain uncertain.
The so-called "threshold" theorem says that, once the error rate per qubit per gate is below a certain value, indefinitely long quantum computation becomes feasible, even if all of the qubits involved are subject to relaxation processes, and all the manipulations with qubits are not exact. The purpose of this article, intended for physicists, is to outline the ideas of quantum error correction and to take a look at the proposed technical instruction for fault-tolerant quantum computation. It seems that the mathematics behind the threshold theorem is somewhat detached from the physical reality, and that some ideal elements are always present in the construction. This raises serious doubts about the possibility of large scale quantum computations, even as a matter of principle.
This is a brief review of the experimental and theoretical quantum computing. The hopes for eventually building a useful quantum computer rely entirely on the so-called "threshold theorem". In turn, this theorem is based on a number of assumptions, treated as axioms, i.e. as being satisfied exactly. Since in reality this is not possible, the prospects of scalable quantum computing will remain uncertain until the required precision, with which these assumptions should be approached, is established. Some related sociological aspects are also discussed. .
The quantum computer is supposed to process information by applying unitary transformations to 2 N complex amplitudes defining the state of N qubits. A useful machine needing N~10 3 or more, the number of continuous parameters describing the state of a quantum computer at any given moment is at least 2 1000 ~10 300 which is much greater than the number of protons in the Universe. However, the theorists believe that the feasibility of large-scale quantum computing has been proved via the “threshold theorem”. Like for any theorem, the proof is based on a number of assumptions considered as axioms. However, in the physical world none of these assumptions can be fulfilled exactly. Any assumption can be only approached with some limited precision. So, the rather meaningless “error per qubit per gate” threshold must be supplemented by a list of the precisions with which all assumptions behind the threshold theorem should hold. Such a list still does not exist. The theory also seems to ignore the undesired free evolution of the quantum computer caused by the energy differences of quantum states entering any given superposition. Another important point is that the hypothetical quantum computer will be a system of 10 3 –10 6 qubits PLUS an extremely complex and monstrously sophisticated classical apparatus. This huge and strongly nonlinear system will generally exhibit instabilities and chaotic behavior.
The Achilles' heel of quantum information processors is the fragility of quantum states and processes. Without a method to control imperfection and imprecision of quantum devices, the probability that a quantum computation succeed will decrease exponentially in the number of gates it requires. In the last fifteen years, building on the discovery of quantum error correction, accuracy threshold theorems were proved showing that error can be controlled using a reasonable amount of resources as long as the error rate is smaller than a certain threshold. We thus have a scalable theory describing how to control quantum systems. The next step is to turn this theory into practice. I will give an overview of some of the progress towards the implmentation of quantum error correction around the world with a focus on results since the last Quantum Error Correction conference at USC. I will compare the various achievements and point towards what still need to be done to get robust quantum information processors.
The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 10–6, would be a formidable factoring engine. Even a smaller less–accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence and Decoherence, 15 to 18 December 1996.
In Chapters 2-5 we gave a detailed presentation of the theory of quantum error correction and fault-tolerant quantum computing. The stage is finally set for a proof of the accuracy threshold theorem [2-9]. As pointed out in Chapter 1, this theorem spells out the conditions under which quantum computing can be done reliably using imperfect quantum gates and in the presence of noise. The proof presented in this chapter follows Ref. [6]. Section 6.1 addresses a number of preliminary topics. It explains why a concatenated quantum error correcting code (QECC) should be used to protect the computational data when fault-tolerance is an issue; it states the principal assumptions underlying the threshold calculations presented in Section 6.2; and finally, it closes with a statement of the accuracy threshold theorem. The proof of this theorem is taken up in Section 6.2. For the error model introduced in Section 6.1, the accuracy threshold is calculated for gates in the Clifford group N(Qn), storage registers, and the Toffoli gate. The calculations make explicit use of the recursive structure of a concatenated QECC and the fault-tolerant procedures introduced in Chapter 5. The theorem is proved by showing that all three cases yield non-zero threshold values.