We define quantum expanders in a natural way. We show that under certain conditions classical expander constructions generalize to the quantum setting, and in particular so does the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of the group PGL. We show that this definition is exactly what is needed for characterizing the complexity of estimating quantum entropies.
We define quantum expanders in a natural way. We show that under certain conditions classical expander constructions generalize to the quantum setting, and in particular so does the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of the group PGL. We show that this definition is exactly what is needed for characterizing the complexity of estimating quantum entropies.
We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2, q) given by Lubotzky et al. (1988). The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold et al. (2000). Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complete (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.
We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.
We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.
For every odd prime power q and D=q+1, we construct an infinite family of Ramanujan quantum expanders of degree D. The construction transfers Morgenstern's Ramanujan Cayley graphs on PSL2(Fp) for p which is an even power of q, through the odd irreducible subrepresentation of the Weil representation of SL2(Fp). For a quantum expander of dimension N, our implementation uses O(log2N) elementary gates, and O(logN) ancilla qudits, using a fixed finite gate set depending on q. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the 2D−1/D singular value bound exactly without any additive error.
We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders. Background. Classical expanders can be defined in either combinatorial or spectral terms, while quantum expanders usually have only a spectral definition. Quantum expanders were introduced in [2] for their application to quantum spin chains and in [5] for applications to quantum statistical zero knowledge. Here we (following [2] and [4]) define a (N, D, λ) quantum expander to be a quantum operation E that • Has N-dimensional input and output. • Has D Kraus operators. • Has second-largest singular value ≤ λ. Equivalently, if E(ρ) = ρ and tr ρσ = 0 then ‖E(σ)‖2 ≤ λ‖σ‖2,
Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the p-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.