Random reversible and quantum circuits form random walks on the alternating group Alt(2n) and unitary group SU(2n), respectively, with each random gate as one step of the walk. Existing bounds on the spectral gap for the t-th moment of these random walks have inverse-polynomial dependence in both n and t. We prove that the gap for random reversible circuits is Ω(n−3) for all t≥1, and the gap for random quantum circuits is Ω(n−3) for t ≤ Θ(2n/2).Importantly, these gaps are independent of t in the respective regimes. We can further improve both gaps to n−1/polylog(n, t) for t ≤ 2Θ(n), which is tight up to polylog factors in n and t. Our spectral gap results have a number of consequences:1)Random reversible circuits with $\mathcal{O}\left( {{n^4}t} \right)$ gates form multiplicative-error t-wise independent (even) permutations for all t ≥ 1; for t ≤ Θ(2n/6.1), we show that $\tilde {\mathcal{O}}\left( {{n^2}t} \right)$ gates suffice.2)Random quantum circuits with $\mathcal{O}\left( {{n^4}t} \right)$ gates form multiplicative-error unitary t-designs for t ≤Θ(2n/2); for t ≤ Θ(22n/5), we show that $\tilde {\mathcal{O}}\left( {{n^2}t} \right)$ gates suffice.3)The robust quantum circuit complexity of random quantum circuits grows linearly for an exponentially long time, proving the robust Brown–Susskind conjecture [1], [2]. We also show an analogous result for random reversible circuits.Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the "PFC ensemble" from [3] together with an expander on the alternating group due to Kassabov [4], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians.

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order$t$. Specifically, a step of the walk on the unitary or orthogonal group of dimension$2^{\mathrm{n}}$is a random Pauli rotation$e^{\mathrm{i}\theta P/2}$. The spectral gap of this random walk is shown to be$\Omega(1/t)$, which coincides with the best previously known bound for a random walk on the permutation group on$\{0,1\}^{\mathrm{n}}$. This implies that the walk gives an$\varepsilon$-approximate unitary t-design in depth$\mathcal{O}(\mathrm{n}t^{2}+t\log\frac{1}{\varepsilon})d$where$d=\mathrm{O}(\log \mathrm{n})$is the circuit depth to implement$e^{\mathrm{i}\theta P/2}$. Our simple proof uses quadratic Casimir operators of Lie algebras.

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using ${\mathcal O}( nT {polylog} (nT/\epsilon))$ gates with depth ${\mathcal O}(T { polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires ${\widetilde{\Omega}}(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To the best of our knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb--Robinson bounds. In the appendix, we prove a Lieb--Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb--Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

We study the problem of learning a Hamiltonian H to precision $\varepsilon$, supposing we are given copies of its Gibbs state $\rho =\exp(-\beta H)/\mathrm{Tr}(\exp(-\beta H))$ at a known inverse temperature $\beta$. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of $\rho$ needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low $\beta$) regime, their algorithm has sample complexity poly (N, 1/$\beta$, 1/$\varepsilon$) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error $\varepsilon$ with sample complexity $S=O(\log N/(\beta\varepsilon)^{2}$) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm’s sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary $e^{-i t H}$ in a small t regime with similar sample and time complexity.

We present a quantum LDPC code family that has distance Ω(N3/5/polylog(N)) and Θ(N3/5) logical qubits, where N is the code length. This is the first quantum LDPC code construction that achieves distance greater than N1/2 polylog(N). The construction is based on generalizing the homological product of codes to a fiber bundle.

We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error ε using O(T polylog(nT/ε)) gates with depth O(T polylog(nT/ε)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/ε. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error ε in trace distance required O(dr2/ε2) copies for a d-dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires O(dr/δ)ln (d/δ) copies to estimate ρ to error δ in infidelity, and a matching lower bound up to logarithmic factors. This implies O((dr/ε2)ln (d/ε)) copies suffice to achieve error ε in trace distance. We also prove that for independent (product) measurements, Ω(dr2/δ2)/ ln(1/δ) copies are necessary in order to achieve error δ in infidelity. For fixed d, our measurement can be implemented on a quantum computer in time polynomial in n.

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. This is the quantum analogue of the problem of estimating a probability distribution given some number of samples.