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arXiv:2409.17971 · International Mathematics Research Notices

Optimal Quantum (Tensor Product) Expanders From Unitary Designs

Cécilia Lancien

Published Dec 30, 2025• 1 citation
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Abstract

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 $2$-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 $U^{\otimes k}$, with $U$ sampled from a $2k$-design measure, then the corresponding random quantum channel is typically an optimal $k$-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).

DOI
10.1093/imrn/rnaf383

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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).
Abstract. In this study, we generate quantum channels with random Kraus operators to typically obtain almost twirling quantum channels and quantum expanders. To prove the concentration phenomena, we use matrix Bernstein’s inequality. In this way, our random models do not utilize Haar-distributed unitary matrices or Gaussian matrices. Rather, as in the preceding research, we use unitary t-designs to generate mixed tensor-product unitary channels acting on Cdt. Although our bounds in Schatten p-norm are valid only for 1 ≤p≤ 2, we show that they are typically quantum ϵ-twirling channels with the tail bound proportional to 1/poly(dt), while such bounds were previously constants. The number of required Kraus operators was also improved by powers of logdandt, to be proportional todtlogd/ϵ2. Such random quantum channels are also typically quantum expanders, but the number of Kraus operators must grow proportionally totlogdin our case. Finally, a new non-unital model of super-operators generated by bounded and isotropic random Kraus operators was introduced, which can be typically rectified to yield almost randomizing quantum channels and quantum expanders.
We introduce the concept of quantum tensor product expanders. These generalize the concept of quantum expanders, which are quantum maps that are efficient randomizers and use only a small number of Kraus operators. Quantum tensor product expanders act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.
In this work, we prove a lower bound on the difference between the first and second singular values of quantum channels induced by random isometries, that is tight in the scaling of the number of Kraus operators. This allows us to give an upper bound on the difference between the first and second largest (in modulus) eigenvalues of random channels with same large input and output dimensions for finite number of Kraus operators k≥169k\geq 169k≥169. Moreover, we show that these random quantum channels are quantum expanders, answering a question posed by Hastings. As an application, we show that ground states of infinite 1D spin chains, which are well-approximated by matrix product states, fulfill a principle of maximum entropy.
We introduce the concept of quantum tensor product expanders. These are expanders that act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.
For large d, we study quantum channels on C d obtained by selecting ran- domly N independent Kraus operators according to a probability measure µ on the unitary group U(d). When µ is the Haar measure, we show that for N d/e 2 , such a channel is e-randomizing with high probability, which means that it maps every state within distance e/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discreti- zation argument. Moreover, for general µ, we obtain an e-randomizing channel provided N d (log d) 6 /e 2 .F ord = 2 k (k qubits), this includes Kraus operators obtained by ten- soring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.
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.