Efficient classical simulation of cluster state quantum circuits with alternative inputs

Sahar Atallah1, Michael Garn1, Sania Jevtic2, Yukuan Tao3, and Shashank Virmani1

1Department of Mathematics, Brunel University London, Kingston Ln, Uxbridge, UB8 3PH, United Kingdom
2Phytoform Labs Ltd., Lawes Open Innovation Hub, West Common, Harpenden, Hertfordshire, England, AL5 2JQ, United Kingdom
3Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, 03755, USA

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We provide new examples of pure entangled systems related to cluster state quantum computation that can be efficiently simulated classically. In cluster state quantum computation input qubits are initialised in the `equator' of the Bloch sphere, $CZ$ gates are applied, and finally the qubits are measured adaptively using $Z$ measurements or measurements of $\cos(\theta)X + \sin(\theta)Y$ operators. We consider what happens when the initialisation step is modified, and show that for lattices of finite degree $D$, there is a constant $\lambda \approx 2.06$ such that if the qubits are prepared in a state that is within $\lambda^{-D}$ in trace distance of a state that is diagonal in the computational basis, then the system can be efficiently simulated classically in the sense of sampling from the output distribution within a desired total variation distance. In the square lattice with $D=4$ for instance, $\lambda^{-D} \approx 0.056$. We develop a coarse grained version of the argument which increases the size of the classically efficient region. In the case of the square lattice of qubits, the size of the classically simulatable region increases in size to at least around $\approx 0.070$, and in fact probably increases to around $\approx 0.1$. The results generalise to a broader family of systems, including qudit systems where the interaction is diagonal in the computational basis and the measurements are either in the computational basis or unbiased to it. Potential readers who only want the short version can get much of the intuition from figures 1 to 3.

An important problem in study of complex quantum systems is the question of when and how quantum systems can be efficiently simulated using conventional classical computers. This question has broad implications. In quantum many-body physics for example, better classical simulation methods can lead to improved numerical simulations as well as new physical insights. Whereas in quantum computing an improved understanding of when quantum systems can or cannot be efficiently simulated classically helps us to elucidate how quantum algorithms can outperform classical ones.

However, in spite of its central importance, any answers we have to this question are far from being complete.

In this work we make progress on this problem by providing a new way to classically simulate a family of pure entangled quantum systems. The family is non-trivial in that that every state within it is pure (i.e. isolated from any environment) and multi-party entangled. For one set of parameters the family contains ideal cluster state quantum computation. However, for other parameter regimes the simulation method is efficient. The fact that these systems can be efficiently simulated classically was previously unknown. Moreover, the method provides a type of local-hidden variable model for some measurements on otherwise pure entangled quantum systems.

The method has interesting connections to the foundations of physics. It works by describing the systems as non-entangled states in a kind of non-quantum theory. In this theory the constituent particles are not described by the usual qubit Bloch sphere, but a state space that is more akin to a cylinder. However, for some of the input states in the family this non-quantum theory breaks down, yielding negative probabilities. Where it does not break down is exactly where it provides an efficient classical simulation.

The method is amenable to a certain form of coarse graining, where particles are grouped into blocks and treated as individual particles. This increases significantly the set of states that can be efficiently simulated classically.

The method can also be generalised to a wider range of systems where particles are first interacted by finite depth commuting circuits and then measured in particular bases.

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Cited by

[1] Sahar Atallah, Michael Garn, Yukuan Tao, and Shashank Virmani, "Classically efficient regimes in measurement based quantum computation performed using diagonal two qubit gates and cluster measurements", arXiv:2307.01800, (2023).

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