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

Classical simulability, entanglement breaking, and quantum computation thresholds (Invited Paper)

S. Virmani, Susana F. Huelga, Martin Bodo Plenio

Published May 23, 2005• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.611662

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Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup $G$ of $U(d)$. In particular we seek $G$ such that $G\otimes G$ has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case $d = 2$ we exhaustively treat all finite irreducible subgroups of $U(2)$ and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the $n^{\rm th}$ root of $Z$ for $n \in \N$.

Quantum Information and Computation•Jan 1, 2008•22 citations
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 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 consider the possibility of adding noise to a quantum circuit to make it efficiently simulatable classically. In previous works, this approach has been used to derive upper bounds to fault tolerance thresholds—usually by identifying a privileged resource, such as an entangling gate or a non-Clifford operation, and then deriving the noise levels required to make it 'unprivileged'. In this work, we consider extensions of this approach where noise is added to Clifford gates too and then 'commuted' around until it concentrates on attacking the non-Clifford resource. While commuting noise around is not always straightforward, we find that easy instances can be identified in popular fault tolerance proposals, thereby enabling sharper upper bounds to be derived in these cases. For instance we find that if we take Knill's (2005 Nature 434 39) fault tolerance proposal together with the ability to prepare any possible state in the XY plane of the Bloch sphere, then not more than 3.69% error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's γ noise model is sufficient. These bounds have been derived without noise being added to the decoding parts of the circuits. Introducing such noise in a toy example suggests that the present approach can be optimized further to yield tighter bounds.
Determining the highest amount of noise that quantum circuits can handle is an interesting and crucial task in the development of fault-tolerant quantum computation. Previous work has constrained this upper limit for local depolarizing noise to ≈45 %\approx 45~\%≈45 % for circuits constructed using the universal Clifford with T gate set by finding the noise threshold where the gate set loses universality. In this work, we use a similar method with the matchgate with non-Gaussian resource universal gate set to find a limit of p=(8−26)/5p=(8 - 2 \sqrt{6})/5p=(8−26​)/5 or ≈62 %\approx 62~\%≈62 % for fermionic quantum computing that is independent of circuit depth. To do this, we use Uhlmann-Wootters concurrences for a 4-mode fermionic Choi state representing a resourceful gate combined with local depolarizing noise to determine when the combined channel is convex Gaussian. These bounds are not directly comparable due to differences in noise model making the bound of ≈62 %\approx 62~\%≈62 % the best known for fermions.
We consider the possibility of adding noise to a quantum circuit to make it efficiently simulatable classically. In previous works, this approach has been used to derive upper bounds to fault tolerance thresholds—usually by identifying a privileged resource, such as an entangling gate or a non-Clifford operation, and then deriving the noise levels required to make it ‘unprivileged’. In this work, we consider extensions of this approach where noise is added to Clifford gates too and then ‘commuted’ around until it concentrates on attacking the non-Clifford resource. While commuting noise around is not always straightforward, we find that easy instances can be identified in popular fault tolerance proposals, thereby enabling sharper upper bounds to be derived in these cases. For instance we find that if we take Knill's (2005 Nature 434 39) fault tolerance proposal together with the ability to prepare any possible state in the XY plane of the Bloch sphere, then not more than 3.69% error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's γ noise model is sufficient. These bounds have been derived without noise being added to the decoding parts of the circuits. Introducing such noise in a toy example suggests that the present approach can be optimized further to yield tighter bounds.
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
Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup GGG of U(d)U(d)U(d). In particular we seek GGG such that G⊗GG\otimes GG⊗G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2d = 2d=2 we exhaustively treat all finite irreducible subgroups of U(2)U(2)U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the nthn^{\rm th}nth root of ZZZ for n∈Nn \in \Nn∈N.