Random unitaries give quantum expanders

Abstract

We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group.Recently, two papers [1, 2] introduced the idea of expander maps: quantum analogues of expander graphs. An expander graph[4] may be defined in several ways. One is the property of having a large number of vertices, a small coordination number for each vertex, and also having a gap in the spectrum of the diffusion equation on the graph, so that a particle classically diffusing on an expander graph rapidly loses memory of where it started. In the quantum case, we replace the random process of diffusion by a completely positive, trace preserving map E(M ). We define a quantum expander to be such a map from the space of N -by-N matrices M to the same space with the following properties. First, N is large, in analogy to the large number of vertices. Second, the map has eigenvalues λ 1 , λ 2 , λ 3 , ..., λ N 2 , with λ 1 = 1 and |λ a | ≤ 1 − δ for all a > 1 so that the eigenvalue spectrum has a gap. Finally, the map can be written asfor some relatively small value of D, with These maps were applied in [1] to construct many-body states in one dimension with the property of having a short correlation length (this corresponds to the gap δ in the spectrum of eigenvalues of E), small Hilbert space dimension on each site (this corresponds to the small D), and yet large entanglement entropy (this corresponds to the large entropy of the eigenvector of E with unit eigenvalue). Since expander graphs have a large number of applications in problems dealing with classical statistics, such as in error-correcting codes [5], derandomization, and the PCP theorem [6], to name a few, it seems worth further exploring the quantum case.A number of possible forms of an expander map are possible: in [2] an expander was defined as havingfor some unitary matrices U (s), and in fact this is the form of A(s) considered in this paper. However, the more general definition with arbitrary A(s) constrained byA(s)A † (s) = 1 1 seems also useful; in fact, although we do not consider it in this paper, it may be useful to weaken this constraint further, and explore the properties of completely positive, trace preserving maps, with no other constraint on the A, requiring only that the entropy of the density matrix which is the eigenvector with unit eigenvalue is large [3].Our goal is to try to find families of maps with arbitrarily large N , such that the gap δ is bounded below by some N -independent constant and such that D does not grow too rapidly with N . The first paper[1] provided an explicit construction of such a family of maps with D of order log(N ) and provided numerical evidence for an alternate construction with D independent ...

Quantum expanders and geometry of operator spaces

“…We refer to [17,12] for more information on expanders and Ramanujan graphs. The next result due to Hastings [11] has been a crucial inspiration for our work: Lemma 1.8 (Hastings). If we equip U (N ) n with its normalized Haar measure P, then for each n and ε > 0 the set R ε (n, N ) defined above satisfies…”

“…The term "Quantum Expander" is used by Hastings in [11] and by Ben-Aroya and Ta-Shma in [2] to designate a sequence…”

“…The term "Quantum Expander" is used by Hastings in [11] and by Ben-Aroya and Ta-Shma in [2] to designate a sequence {U (N ) | N ≥ 1} of n-tuples U (N ) = (U (N ) 1 , · · · , U (N ) n ) of N × N unitary matrices such that there is an ε > 0 satisfying the following "spectral gap" condition:…”

“…This generalizes to operator spaces a classical geometric result on n-dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.The term "Quantum Expander" is used by Hastings in [11] and by Ben-Aroya and Ta-Shma in [2] to designate a sequencen ) of N × N unitary matrices such that there is an ε > 0 satisfying the following "spectral gap" condition:where . 2 denotes the Hilbert-Schmidt norm on M N .…”

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“…Following the publication of this construction (given in [8]), Hastings [20] showed, using elegant techniques, that quantum expanders cannot be better than Ramanujan, i. e., cannot have spectral gap better than 1 − 2 √ D − 1/D. Hastings also showed that taking D random unitaries gives an almostRamanujan expander.…”