arXiv subjects

Yinchen Liu

Publications and source records attributed to Yinchen Liu.

23 records · Page 2Linked to original sources

arXiv2024

On the complexity of sampling from shallow Brownian circuits

While many statistical properties of deep random quantum circuits can be deduced, often rigorously and other times heuristically, by an approximation to global Haar-random unitaries, the statistics of constant-depth random quantum circuits are generally less well-understood due to a lack of amenable tools and techniques. We circumvent this barrier by considering a related constant-time Brownian circuit model which shares many similarities with constant-depth random quantum circuits but crucially allows for direct calculations of higher order moments of its output distribution. Using mean-field (large-n) techniques, we fully characterize the output distributions of Brownian circuits at shallow depths and show that they follow a Porter-Thomas distribution, just like in the case of deep circuits, but with a truncated Hilbert space. The access to higher order moments allows for studying the expected and typical Linear Cross-entropy (XEB) benchmark scores achieved by an ideal quantum computer versus the state-of-the-art classical spoofers for shallow Brownian circuits. We discover that for these circuits, while the quantum computer typically scores within a constant factor of the expected value, the classical spoofer suffers from an exponentially larger variance. Numerical evidence suggests that the same phenomenon also occurs in constant-depth discrete random quantum circuits, like those defined over the all-to-all architecture. We conjecture that the same phenomenon is also true for random brickwork circuits in high enough spatial dimension.

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arXiv2024

Quantum advantage from measurement-induced entanglement in random shallow circuits

We study random constant-depth quantum circuits in a two-dimensional architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value. For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution. Here we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O(d). Finally, we introduce a two-dimensional, depth-2, "coarse-grained" circuit architecture, composed of random Clifford gates acting on O(log n) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.

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arXiv2023

A rapidly mixing Markov chain from any gapped quantum many-body system

We consider the computational task of sampling a bit string $x$ from a distribution $π(x)=|\langle x|ψ\rangle|^2$, where $ψ$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the inverse spectral gap of $H$ and the mixing time of an associated continuous-time Markov Chain with steady state $π$. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes $\langle y|ψ\rangle/\langle x|ψ\rangle$ are efficiently computable, the spectral gap of $H$ is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.

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arXiv2023

Classical simulation of peaked shallow quantum circuits

An $n$-qubit quantum circuit is said to be peaked if it has an output probability that is at least inverse-polynomially large as a function of $n$. We describe a classical algorithm with quasipolynomial runtime $n^{O(\log{n})}$ that approximately samples from the output distribution of a peaked constant-depth circuit. We give even faster algorithms for circuits composed of nearest-neighbor gates on a $D$-dimensional grid of qubits, with polynomial runtime $n^{O(1)}$ if $D=2$ and almost-polynomial runtime $n^{O(\log{\log{n}})}$ for $D>2$. Our sampling algorithms can be used to estimate output probabilities of shallow circuits to within a given inverse-polynomial additive error, improving previously known methods. As a simple application, we obtain a quasipolynomial algorithm to estimate the magnitude of the expected value of any Pauli observable in the output state of a shallow circuit (which may or may not be peaked). This is a dramatic improvement over the prior state-of-the-art algorithm which had an exponential scaling in $\sqrt{n}$.

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arXiv2022

How to simulate quantum measurement without computing marginals

We describe and analyze algorithms for classically simulating measurement of an $n$-qubit quantum state $ψ$ in the standard basis, that is, sampling a bit string $x$ from the probability distribution $|\langle x|ψ\rangle|^2$. Our algorithms reduce the sampling task to computing poly$(n)$ amplitudes of $n$-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. First we consider the case where $|ψ\rangle=U|0^n\rangle$ is the output state of an $m$-gate quantum circuit $U$. We propose an exact sampling algorithm which involves computing $O(m)$ amplitudes of $n$-qubit states generated by subcircuits of $U$ spanned by the first $t=1,2,\ldots,m$ gates. We show that our algorithm can significantly accelerate quantum circuit simulations based on tensor network contraction methods or low-rank stabilizer decompositions. As another striking consequence we obtain an efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph, generalizing a previous algorithm which was known to be efficient only under restrictive topological constraints on the ordering of single-qubit measurements. Second, we consider the case in which $ψ$ is the unique ground state of a local Hamiltonian with a spectral gap that is lower bounded by an inverse polynomial function of $n$. We prove that a simple Metropolis-Hastings Markov Chain mixes rapidly to the desired probability distribution provided that $ψ$ obeys a certain technical condition, which we show is satisfied for all sign-problem free Hamiltonians. This gives a sampling algorithm which involves computing $\mathrm{poly}(n)$ amplitudes of $ψ$.

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