We develop a machine learning method to construct accurate ground-state wave functions of strongly interacting and entangled quantum spin as well as fermionic models on lattices. A restricted Boltzmann machine algorithm in the form of an artificial neural network is combined with a conventional variational Monte Carlo method with pair product (geminal) wave functions and quantum number projections. The combination allows an application of the machine learning scheme to interacting fermionic systems. The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.
The surface code, with a simple modification, exhibits ultrahigh error-correction thresholds when the noise is biased toward dephasing. Here, we identify features of the surface code responsible for these ultrahigh thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is 50%, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial-time decoding algorithm. We demonstrate that the subthreshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough and smooth open boundaries, it is controlled by the parameter g = gcd(j, k), where j and k are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for coprime codes that have g = 1, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: The same logical failure rate achievable with a square surface code and n physical qubits can be obtained with a coprime or rotated surface code using only O( √ n) physical qubits. Finally, we use approximate maximum-likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased toward dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.
The surface code is a many-body quantum system, and simulating it in generic conditions is computationally hard. While the surface code is believed to have a high threshold, the numerical simulations used to establish this threshold are based on simplified noise models. We present a tensor-network algorithm for simulating error correction with the surface code under arbitrary local noise. We use this algorithm to study the threshold and the subthreshold behavior of the amplitudedamping and systematic rotation channels. We also compare these results to those obtained by making standard approximations to the noise models.Introduction. -The working principle behind quantum error correction is to "fight entanglement with entanglement," i.e., protect the data against local interaction with the environment by encoding them into delocalized degrees of freedom of a many-body system. Thus, characterizing a fault-tolerant scheme is ultimately a problem of quantum many-body physics.While simulating quantum many-body systems is generically hard, particular systems have additional structure that can be taken advantage of. For example, free-fermion Hamiltonians have algebraic properties that make them exactly solvable. An analogy in stabilizer quantum error correction is Pauli noise, where errors are Pauli operators drawn from some fixed distribution. Because of their algebraic structure, Pauli noise models can be simulated efficiently using the stabilizer formalism [1]. Beyond Pauli noise, noise composed of Clifford gates and projections onto Pauli eigenstates can also be simulated efficiently using the same methods [2].While such efficiently simulable noise models can be useful to benchmark fault-tolerant schemes, they do not represent most models of practical interest. For instance, qubits that are built out of nondegenerate energy eigenstates are often subject to relaxation, a.k.a. amplitude damping. Miscalibrations often result in systematic errors corresponding to small unitary rotations [3]. Given that these processes do not have efficient descriptions within the stabilizer formalism, understanding how a given fault-tolerant scheme will respond to them is a difficult and important problem.The simplest approach to such many-body problems is brute-force simulation, where an arbitrary state in Hilbert space is represented as an exponentially large vector of coefficients. Using such methods, small surface codes (up to distance 3) have been simulated under nonClifford noise [4]. In another study, brute-force simulation of the seven-qubit Steane code was performed without concatenation [5]. Simulation of such low distance codes allows comparison of noise at the logical level to the noise on the physical level; however, it is difficult to infer quantities of interest such as thresholds or overheads from such small simulations. Another approach, akin to the use of tight-binding approximations in solid-state physics, is to approximate these noise processes with efficiently simulable ones [2,6]. However, the accuracy of these ap...
Abstract. Recently, it was shown that the non-local correlations needed for measurement-based quantum computation (MBQC) can be revealed in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model involving nearest-neighbour spin-3/2 interactions on a honeycomb lattice. This state is not singular but resides in the disordered phase of the ground states of a large family of Hamiltonians characterized by short-range-correlated valence bond solid states. By applying local filtering and adaptive single-particle measurements, we show that most states in the disordered phase can be reduced to a graph of correlated qubits that is a scalable resource for MBQC. At the transition between the disordered and Néel ordered phases, we find a transition from universal to non-universal states as witnessed by the scaling of percolation in the reduced graph state.
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