Search

We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations, single- and two-qubit operations, as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an $L^1$-norm approximation to the state. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|\mathsf{GKP}_{\kappa,\Delta}\rangle$. Here $\kappa^{-2}$ is the variance of the envelope and $\Delta^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and -- conditioned on acceptance -- the output state is polynomially close in $(\kappa,\Delta)$ to the state $|\mathsf{GKP}_{\kappa,\Delta}\rangle$. The size of our circuit is linear in $(\log 1/\kappa,\log 1/\Delta)$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states., Comment: 85 pages, 12 figures

We propose efficient classical algorithms which (strongly) simulate the action of bosonic linear optics circuits applied to superpositions of Gaussian states. Our approach relies on an augmented covariance matrix formalism to keep track of relative phases between individual terms in a linear combination. This yields an exact simulation algorithm whose runtime is polynomial in the number of modes and the size of the circuit, and quadratic in the number of terms in the superposition. We also present a faster approximate randomized algorithm whose runtime is linear in this number. Our main building blocks are a formula for the triple overlap of three Gaussian states and a fast algorithm for estimating the norm of a superposition of Gaussian states up to a multiplicative error. Our construction borrows from earlier work on simulating quantum circuits in finite-dimensional settings, including, in particular, fermionic linear optics with non-Gaussian initial states and Clifford computations with non-stabilizer initial states. It provides algorithmic access to a practically relevant family of non-Gaussian bosonic circuits., Comment: 43 pages and 11 figures

We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits: To execute an arbitrary quantum circuit on $n$ qubits, we give a 3D quantum fault-tolerance architecture involving $O(n^{3/2} \log^3 n)$ qubits, and a quasi-2D architecture using $O(n^2 \log^3 n)$ qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead and a quasi-polylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing., Comment: 38 pages, 4 figures

We present a computational problem with the following properties: (i) Every instance can be solved with near-certainty by a constant-depth quantum circuit using only nearest-neighbor gates in 3D even when its implementation is corrupted by noise. (ii) Any constant-depth classical circuit composed of unbounded fan-in AND, OR, as well as NOT gates, i.e., an AC0-circuit, of size smaller than a certain subexponential, fails to solve a uniformly random instance with probability greater than a certain constant. Such an advantage against unbounded fan-in classical circuits was previously only known in the noise-free case or without locality constraints. We overcome these limitations, proposing a quantum advantage demonstration amenable to experimental realizations. Subexponential circuit-complexity lower bounds have traditionally been referred to as exponential. We use the term colossal since our fault-tolerant 3D architecture resembles a certain Roman monument., Comment: 44 pages, 12 figures

Local update recovery seeks to maintain quantum information by applying local correction maps alternating with and compensating for the action of noise. Motivated by recent constructions based on quantum LDPC codes in the finite-dimensional setting, we establish an analytic upper bound on the fault-tolerance threshold for concatenated GKP-stabilizer codes with local update recovery. Our bound applies to noise channels that are tensor products of one-mode beamsplitters with arbitrary environment states, capturing, in particular, photon loss occurring independently in each mode. It shows that for loss rates above a threshold given explicitly as a function of the locality of the recovery maps, encoded information is lost at an exponential rate. This extends an early result by Razborov from discrete to continuous variable (CV) quantum systems. To prove our result, we study a metric on bosonic states akin to the Wasserstein distance between two CV density functions, which we call the bosonic Wasserstein distance. It can be thought of as a CV extension of a quantum Wasserstein distance of order 1 recently introduced by De Palma et al. in the context of qudit systems, in the sense that it captures the notion of locality in a CV setting. We establish several basic properties, including a relation to the trace distance and diameter bounds for states with finite average photon number. We then study its contraction properties under quantum channels, including tensorization, locality and strict contraction under beamsplitter-type noise channels. Due to the simplicity of its formulation, and the established wide applicability of its finite-dimensional counterpart, we believe that the bosonic Wasserstein distance will become a versatile tool in the study of CV quantum systems., Comment: 30 pages, 2 figures

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity., Comment: 68 pages and 15 figures; updated to match published version

Gate-teleportation circuits are arguably among the most basic examples of computations believed to provide a quantum computational advantage: In seminal work [Quantum Inf. Comput., 4(2):134--145], Terhal and DiVincenzo have shown that these circuits elude simulation by efficient classical algorithms under plausible complexity-theoretic assumptions. Here we consider possibilistic simulation [Phys. Rev. A 106, 062430 (2022)], a particularly weak form of this task where the goal is to output any string appearing with non-zero probability in the output distribution of the circuit. We show that even for single-qubit Clifford-gate-teleportation circuits this simulation problem cannot be solved by constant-depth classical circuits with bounded fan-in gates. Our results are unconditional and are obtained by a reduction to the problem of computing the parity, a well-studied problem in classical circuit complexity., Comment: 31 pages 10 figures. Typos fixed. New Section 3.3.3 Other word problems for Clifford modulo Pauli groups

We propose a scheme for fault-tolerant long-range entanglement generation at the ends of a rectangular array of qubits of length $R$ and a square cross section of size $d\times d$ with $d=O(\log R)$. Up to an efficiently computable Pauli correction, the scheme generates a maximally entangled state of two qubits using a depth-$6$ circuit consisting of nearest-neighbor Clifford gates and local measurements only. Compared with existing fault-tolerance schemes for quantum communication, the protocol is distinguished by its low latency: starting from a product state, the entangled state is prepared in a time $O(t_{\textrm{local}})$ determined only by the local gate and measurement operation time $t_{\textrm{local}}$. Furthermore, the requirements on local repeater stations are minimal: Each repeater uses only $\Theta(\log^2 R)$ qubits with a lifetime of order $O(t_{\textrm{local}})$. We prove a converse bound $\Omega(\log R)$ on the number of qubits per repeater among all low-latency schemes for fault-tolerant quantum communication over distance $R$. Furthermore, all operations within a repeater are local when the qubits are arranged in a square lattice. The noise-resilience of our scheme relies on the fault-tolerance properties of the underlying cluster state. We give a full error analysis, establishing a fault-tolerance threshold against general (circuit-level) local stochastic noise affecting preparation, entangling operations and measurements. This includes, in particular, errors correlated in time and space. Our conservative analytical estimates are surprisingly optimistic, suggesting that the scheme is suited for long-range entanglement generation both in and between near-term quantum computing devices., Comment: 68 pages, 11 figures

We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group $G$. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-$1$ circuit composed of generalized Pauli gates., Comment: 48 pages, 18 figures. Revisions in v2: proof of Theorem 4.1 is extended to all solvable groups by lifting an implicit assumption about the group structure

Proposed hybrid algorithms encode a combinatorial cost function into a problem Hamiltonian and optimize its energy by varying over a set of states with low circuit complexity. Classical processing is typically only used for the choice of variational parameters following gradient descent. As a consequence, these approaches are limited by the descriptive power of the associated states. We argue that for certain combinatorial optimization problems, such algorithms can be hybridized further, thus harnessing the power of efficient non-local classical processing. Specifically, we consider combining a quantum variational ansatz with a greedy classical post-processing procedure for the MaxCut-problem on $3$-regular graphs. We show that the average cut-size produced by this method can be quantified in terms of the energy of a modified problem Hamiltonian. This motivates the consideration of an improved algorithm which variationally optimizes the energy of the modified Hamiltonian. We call this a twisted hybrid algorithm since the additional classical processing step is combined with a different choice of variational parameters. We exemplify the viability of this method using the quantum approximate optimization algorithm (QAOA), giving analytic lower bounds on the expected approximation ratios achieved by twisted QAOA. These show that the necessary non-locality of the quantum ansatz can be reduced compared to the original QAOA: We find that for levels $p=2,\ldots, 6$, the level $p$ can be reduced by one while roughly maintaining the expected approximation ratio. This reduces the circuit depth by $4$ and the number of variational parameters by $2$., Comment: 29 pages, 12 figures