Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place.

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.

Preparing encoded logical states is the first step in a fault-tolerant quantum computation. Standard approaches based on concatenation or repeated measurement incur a significant time overhead. The Raussendorf-Bravyi-Harrington cluster state offers an alternative: a single-shot preparation of encoded states of the surface code, by means of a constant depth quantum circuit, followed by a single round of measurement and classical feedforward. In this work we generalize this approach and prove that single-shot logical state preparation can be achieved for arbitrary quantum LDPC codes. Our proof relies on a minimum-weight decoder and is based on a generalization of Gottesman's clustering-of-errors argument. As an application, we also prove single-shot preparation of the encoded GHZ state in arbitrary quantum LDPC codes. This shows that adaptive noisy constant depth quantum circuits are capable of generating generic robust long-range entanglement.

A quantum system coupled to a bath at some fixed, finite temperature converges to its Gibbs state. This thermalization process defines a natural, physically-motivated model of quantum computation. However, whether quantum computational advantage can be achieved within this realistic physical setup has remained open, due to the challenge of finding systems that thermalize quickly, but are classically intractable. Here we consider sampling from the measurement outcome distribution of quantum Gibbs states at constant temperatures, and prove that this task demonstrates quantum computational advantage. We design a family of commuting local Hamiltonians (parent Hamiltonians of shallow quantum circuits) and prove that they rapidly converge to their Gibbs states under the standard physical model of thermalization (as a continuous-time quantum Markov chain). On the other hand, we show that no polynomial time classical algorithm can sample from the measurement outcome distribution by reducing to the classical hardness of sampling from noiseless shallow quantum circuits. The key step in the reduction is constructing a fault-tolerance scheme for shallow IQP circuits against input noise.

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.

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U. We also provide a polynomial-time classical algorithm for learning the description of any unknown n-qubit state | ψ ⟩ = U | 0n ⟩ prepared by a shallow quantum circuit U (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of | ψ ⟩. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

We give a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit in the regime of anti-concentration to within inverse polynomial total variation distance. The algorithm is based on a quantum analog of noise induced low degree approximations of Boolean functions, which takes the form of the truncation of a Feynman path integral in the Pauli basis.

As small quantum computers are becoming available on different physical platforms, a benchmarking task known as cross-platform verification has been proposed that aims to estimate the fidelity of states prepared on two quantum computers. This task is fundamentally distributed, as no quantum communication can be performed between the two physical platforms due to hardware constraints, which prohibits a joint SWAP test. In this paper we settle the sample complexity of this task across all measurement and communication settings. The essence of the task, which we call distributed quantum inner product estimation, involves two players Alice and Bob who have k copies of unknown states ρ,σ (acting on ℂd) respectively. Their goal is to estimate Tr(ρσ) up to additive error ε∈(0,1), using local quantum operations and classical communication. In the weakest setting where only non-adaptive single-copy measurements and simultaneous message passing are allowed, we show that k=O(max{1/ε2,√d/ε}) copies suffice. This achieves a savings compared to full tomography which takes Ω(d3) copies with single-copy measurements. Surprisingly, we also show that the sample complexity must be at least Ω(max{1/ε2,√d/ε}), even in the strongest setting where adaptive multi-copy measurements and arbitrary rounds of communication are allowed. This shows that the success achieved by shadow tomography, for sample-efficiently learning the properties of a single system, cannot be generalized to the distributed setting. Furthermore, the fact that the sample complexity remains the same with single and multi-copy measurements contrasts with single system quantum property testing, which often demonstrate exponential separations in sample complexity with single and multi-copy measurements.

Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity in these highly noisy devices. Surprisingly, we show that it remains hard to compute the output probabilities of noisy random quantum circuits without error correction. More formally, so long as the noise rate of the device is below the error detection threshold, we show it is #P-hard to compute the output probabilities of random circuits with a constant rate of noise per gate. This hardness persists even though these probabilities are exponentially close to uniform. Therefore the small deviations away from uniformity are hard to compute, formalizing an important intuition behind Google's supremacy claim. Interestingly these hardness results also have implications for the complexity of experiments in a low-noise setting. The issue here is that prior hardness results for computing output proba-bilities of random circuits are not robust enough to imprecision to connect with the Stockmeyer argument for hardness of sampling from circuits with constant fidelity. We exponentially improve the robustness of prior results to imprecision, both in the cases of Random Circuit Sampling and BosonSampling. In the latter case we bring the proven hardness within a constant factor in the exponent of the robustness required for hardness of sampling for the first time. We then show that our results are in tension with one another - the high-noise result implies the low-noise result is essentially optimal, even with generalizations of our techniques.

The resource framework of quantum coherence was introduced by Baumgratz, Cramer, and Plenio [Phys. Rev. Lett. 113, 140401 (2014)] and further developed by Winter and Yang [Phys. Rev. Lett. 116, 120404 (2016)]. We consider the one-shot problem of distilling pure coherence from a single instance of a given resource state. Specifically, we determine the distillable coherence with a given fidelity under incoherent operations (IO) through a generalization of the Winter-Yang protocol. This is compared to the distillable coherence under maximal incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO), which can be cast as a semidefinite programme, that has been presented previously by Regula et al. [Phys. Rev. Lett. 121, 010401 (2018)]. Our results are given in terms of a smoothed min-relative entropy distance from the incoherent set of states, and a variant of the hypothesis-testing relative entropy distance, respectively. The one-shot distillable coherence is also related to one-shot randomness extraction. Moreover, from the one-shot formulas under IO, MIO, and DIO, we can recover the optimal distillable rate in the many-copy asymptotics, yielding the relative entropy of coherence. These results can be compared with previous work by some of the present authors [Zhao et al., Phys. Rev. Lett. 120, 070403 (2018)] on one-shot coherence formation under IO, MIO, DIO and also SIO. This shows that the amount of distillable coherence is essentially the same for IO, DIO, and MIO, despite the fact that the three classes of operations are very different. We also relate the distillable coherence under strictly incoherent operations (SIO) to a constrained hypothesis testing problem and explicitly show the existence of bound coherence under SIO in the asymptotic regime.