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arXiv:quant-ph/0301108 · Physical Review A

Robustness of quantum gates in the presence of noise

Aram W. Harrow, Michael A. Nielsen

Published Jul 11, 2003• 164 citations
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Abstract

We define several quantitative measures of the robustness of a quantum gate against noise. Exact analytic expressions for the robustness against depolarizing noise are obtained for all bipartite unitary quantum gates, and it is found that the controlled-NOT gate is the most robust two-qubit quantum gate, in the sense that it is the quantum gate which can tolerate the most depolarizing noise and still generate entanglement. Our results enable us to place several analytic upper bounds on the value of the threshold for quantum computation, with the best bound in the most pessimistic error model being ${p}_{\mathrm{th}}<~0.5.$

DOI
10.1103/physreva.68.012308

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We define several quantitative measures of the robustness of a quantum gate against noise. Exact analytic expressions for the robustness against depolarizing noise are obtained for all bipartite unitary quantum gates, and it is found that the controlled-NOT gate is the most robust two-qubit quantum gate, in the sense that it is the quantum gate which can tolerate the most depolarizing noise and still generate entanglement. Our results enable us to place several analytic upper bounds on the value of the threshold for quantum computation, with the best bound in the most pessimistic error model being pth< 0.5.{p}_{\mathrm{th}}<~0.5.pth​< 0.5.
We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary kkk-qubit gates each of whose input wires is subject to depolarizing noise of strength ppp, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1−Θ(1/k)p>1-\Theta(1/\sqrt{k})p>1−Θ(1/k​), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2k=2k=2, our bound is p>35.7%p>35.7\%p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%29.3\%29.3%. These bounds on ppp are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.
We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1-Θ(1/\sqrt{k}), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2, our bound is p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds.
We investigate the amount of noise required to turn a universal quantum gate set into one that can be efficiently modeled classically. This question is useful for providing upper bounds on fault-tolerant thresholds, and for understanding the nature of the quantum-classical computational transition. We refine some previously known upper bounds using two different strategies. The first one involves the introduction of bientangling operations, a class of classically simulable machines that can generate at most bipartite entanglement. Using this class we show that it is possible to sharpen previously obtained upper bounds in certain cases. As an example, we show that under depolarizing noise on the controlled-NOT gate, the previously known upper bound of 74% can be sharpened to around 67%. Another interesting consequence is that measurement-based schemes cannot work using only two-qubit nondegenerate projections. In the second strand of the work we utilize the Gottesman-Knill theorem on the classically efficient simulation of Clifford group operations. The bounds attained using this approach for the $\ensuremath{\pi}∕8$ gate can be as low as 15% for general single-gate noise, and 30% for dephasing noise.
We investigate the amount of noise required to turn a universal quantum gate set into one that can be efficiently modelled classically. This question is useful for providing upper bounds on fault tolerant thresholds, and for understanding the nature of the quantum/classical computational transition. We refine some previously known upper bounds using two different strategies. The first one involves the introduction of bi-entangling operations, a class of classically simulatable machines that can generate at most bipartite entanglement. Using this class we show that it is possible to sharpen previously obtained upper bounds in certain cases. As an example, we show that under depolarizing noise on the controlled-not gate, the previously known upper bound of 74% can be sharpened to around 67%. Another interesting consequence is that measurement based schemes cannot work using only 2-qubit non-degenerate projections. In the second strand of the work we utilize the Gottesman-Knill theorem on the classically efficient simulation of Clifford group operations. The bounds attained using this approach for the pi/8-gate can be as low as 15% for general single gate noise, and 30% for dephasing noise.
We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of thetas = (6 - 2radic2)/7 ap 45%, thereby improving on a previous bound of 50% (due to Razborov, 2004). More precisely, the circuit model for which we prove this bound contains perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional one-qubit gates that are subject to depolarizing noise thetas. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows
We study how much noise can be tolerated by a universal gate set before it loses its quantum-computational power. Specifically we look at circuits with perfect stabilizer operations in addition to imperfect nonstabilizer gates. We prove that for all unitary single-qubit gates there exists a tight depolarizing noise threshold that determines whether the gate enables universal quantum computation or if the gate can be simulated by a mixture of Clifford gates. This exact threshold is determined by the Clifford polytope spanned by the 24 single-qubit Clifford gates. The result is in contrast to the situation wherein nonstabilizer qubit states are used; the thresholds in that case are not currently known to be tight.