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arXiv:2406.10029

Non-Hermitian expander obtained with Haar distributed unitaries

Sarah Timhadjelt

Published Jun 14, 2024• 0 citations
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Abstract

We consider a random quantum channel obtained by taking a selection of $d$ independent and Haar distributed $N$ dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for $d$-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.

DOI
10.48550/arxiv.2406.10029

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We consider a random quantum channel obtained by taking a selection of ddd independent and Haar distributed NNN dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for ddd-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.
We consider a random quantum channel obtained by taking a selection of [Formula: see text] independent and Haar distributed [Formula: see text]-dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values [7, 8]. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon–Boppana bound for [Formula: see text]-regular graphs. The upper bound is obtained using Schwinger–Dyson equations.
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Abstract In this work we investigate how quantum expanders (i.e., quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a 222-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form U⊗kU^{\otimes k}U⊗k, with UUU sampled from a 2k2k2k-design measure, then the corresponding random quantum channel is typically an optimal kkk-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).
With unitary coin operator that is a random matrix drawn from a uniform distribution with respect to the Haar measure, we construct a variant of a discrete quantum random walk using a d-regular undirected connected simple graph. With each step of the walk, the coin operator random matrix is drawn independently from the distribution. The expectation value over the distribution of random unitaries for the associated quantum channel gives a depolarization channel for the coin subspace and resembles the associated classical random walk where the direction a walker steps depends upon the outcome of a fair coin or balanced d-sided dice. Remarkably, despite the fact that the averaged channel depolarizes the coin subspace, measurements in the vertex subspace can be designed that would reveal information about the initial quantum state, even after many iterations of the channel. We illustrate with examples of quantum walks on Cayley graphs of Abelian groups, such as the cycle and hypercube graphs. For Cayley graphs of Abelian groups, the averaged channel is dephasing in the Fourier basis. These quantum walks are examples of bipartite strongly interacting systems, where one subsystem is strongly perturbed, yet information about the initial state in the other subsystem is potentially measurable forever. Due to its decoherence, a quantum random walk with a Haar-random coin would not be useful in search algorithms but could aid in understanding quantum systems with strongly perturbed subsystems.
We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2n)\mathsf{SU}(2^n)SU(2n) on nnn qubits. (i) Random Pauli Rotation: choose an nnn-qubit Pauli operator PPP and an angle θ∈[0,2π)θ\in [0,2π)θ∈[0,2π), both uniformly at random, and apply eiθPe^{\mathrm i θP}eiθP. (ii) Brickwork Random Unitary Circuit: choose n−1n-1n−1 unitaries UiU_{i}Ui​ uniformly at random from SU(4)\mathsf{SU}(4)SU(4) (including central phases of the parent group) independently, and apply U2j−1U_{2j-1}U2j−1​ on two qubits 2j−1,2j2j-1, 2j2j−1,2j and then U2jU_{2j}U2j​ on two qubits 2j,2j+12j, 2j+12j,2j+1. Importantly, the gap bounds are independent of nnn and apply for all finite dimensional unitary representations of SU(2n)\mathsf{SU}(2^n)SU(2n) uniformly. Our results immediately imply that the Random Pauli Rotation and Brickwork Random Unitary Circuit converge to approximate unitary ttt-designs after O(nt)\mathcal{O}(nt)O(nt) steps. As a consequence, we improve implications for the complexity growth of random unitary circuits as a function of the depth. The existing argument can be applied almost verbatim, but previously the depth had to be O(2cn)\mathcal{O}(2^{cn})O(2cn) for some c<1c < 1c<1. Our unconditional constant spectral gap removes this restriction. We prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit. The former implies convergence to approximate orthogonal ttt-designs. In this case, the implied mixing time with respect to a constant additive and multiplicative error is optimal (up to an absolute constant factor) for t=2t=2t=2.