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arXiv:1310.0938 · The British Journal for the Philosophy of Science

On the Significance of the Gottesman–Knill Theorem

Michael E. Cuffaro

Published Jul 1, 2015• 19 citations

Abstract

According to the Gottesman–Knill theorem, quantum algorithms that utilize only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this article that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a classical computer) is, on reflection, already evident when we consider Bell’s and related inequalities in the context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman–Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman–Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential. 1 Introduction 2 The Gottesman–Knill Theorem 3 The Significance of the Gottesman–Knill Theorem 4 Classical Communication and Locally Causal Description 5 Constraining Locally Causal Descriptions in the Practical Context of Inquiry 6 The Sufficiency of Entanglement Thesis 7 Summary Appendix A Recovering Statistics for Bipartite Entangled States Generated from Gottesman–Knill Operations Appendix B Recovering Statistics of Pauli Measurements on the GHZ State

DOI: 10.1093/bjps/axv016

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According to the Gottesman–Knill theorem, quantum algorithms that utilize only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this article that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a classical computer) is, on reflection, already evident when we consider Bell’s and related inequalities in the context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman–Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman–Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential. 1 Introduction 2 The Gottesman–Knill Theorem 3 The Significance of the Gottesman–Knill Theorem 4 Classical Communication and Locally Causal Description 5 Constraining Locally Causal Descriptions in the Practical Context of Inquiry 6 The Sufficiency of Entanglement Thesis 7 Summary Appendix A Recovering Statistics for Bipartite Entangled States Generated from Gottesman–Knill Operations Appendix B Recovering Statistics of Pauli Measurements on the GHZ State

According to the Gottesman-Knill theorem, any quantum algorithm utilising operations chosen exclusively from a particular restricted set are efficiently simulable by a classical computer. Since some of these algorithms involve entangled states, it is commonly concluded that entanglement is insufficient to enable quantum speedup. As I explain, however, the operations belonging to this set are precisely those which will never yield a violation of the Bell inequalities. Thus it should be no surprise that entangled quantum states which only undergo operations in this set are efficiently simulable classically. What the Gottesman-Knill theorem shows us is that it is possible to use an entangled state to less than its full potential. Nevertheless, there is a meaningful sense in which entanglement is sufficient for quantum speedup: an entangled quantum state provides sufficient physical resources to enable quantum speedup, whether or not one elects to use these resources fully.

Restricted models of quantum computation are ones that have less power than a universal quantum computer. We studied the consequences of removing particular properties from a universal quantum computer to discover whether those resources were important.In the first part of the thesis we studied universal quantum computers which are implemented using Clifford gates, adaptive measurements, and magic states. The Gottesman–Knill theorem shows that circuits in this form which do not use magic states can be simulated by a classical computer. We extended this result to show that all circuits in this form can be partially simulated; the same computation can be implemented using a smaller quantum computer with the assistance of some polynomial time classical computation. We also identified a subclass of these computations that can be shown to not be entirely classically simulated by any method, given certain complexity theoretic assumptions are true.In the next part of the thesis we examine the role of entanglement in noisy quantum computations. Entanglement is necessary for noiseless quantum computers to have any quantum advantage, but it is not known whether the same is true for mixed state quantum computers. We show that entanglement, unexpectedly, does play a crucial role in the most well known mixed state computer: the one clean qubit model.Finally, we investigate how closely classical simulation is related to another idea of classicality.This notion captures how easily the final state of a computation can be learnt, given samples of measurements from it. We find an extra condition under which a circuit that is classically simulable is also efficiently learnable.

The Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation. Quanta 2024; 13: 20–27.

We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.

We present a local hidden-variable model supplemented by classical communication that reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an n-qubit GHZ state (or "cat state"). The simulation is efficient since the required amount of communication scales linearly with the number of qubits, even though there are Bell-type inequalities for these states for which the amount of violation grows exponentially with n. The structure of our model yields insight into the Gottesman-Knill theorem by demonstrating that, at least in this limited case, the correlations in the set of nonlocal hidden variables represented by the stabilizer generators are captured by an appropriate set of local hidden variables augmented by n-2 bits of classical communication.

Quantum systems are known to offer advantages over their classical counterpart in communication complexity protocols, where the aim is to minimize the amount of information exchange between distant parties to compute global functions of their distributed inputs. In this work, we establish that any one-way communication protocol implemented using a prime-dimensional quantum system -- restricted to stabilizer-state encodings and Clifford-operation decodings -- can be exactly simulated by transmitting a classical system of the same dimension, given access to shared randomness between the sender and receiver. In direct analogy with the Gottesman-Knill theorem, which attributes quantum computational speedup to non-stabilizer resources, commonly known as the magic resources, our result identifies the same non-stabilizer resources as the essential ingredient for the quantum advantage in one-way communication complexity. Furthermore, we present explicit tasks where even a 'minimal magic resource' suffices to achieve a provable quantum advantage, highlighting its efficient use in communication protocols.