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The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size $L$, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has $O(L^2)$ time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly $2L$ time steps to encode an unknown state in a distance $L$ planar code. We further show how an $O(L)$ complexity local unitary encoder for the toric code can be found by enforcing locality in the $O(log L)$-depth non-local renormalisation encoder. We relate these techniques by providing an $O(L)$ local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.
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We introduce a scheme for efficiently describing pure states of strongly correlated fermions in higher dimensions using unitary circuits featuring a causal cone. A local way of computing local expectation values is presented. We formulate a dynamical reordering scheme, corresponding to time-adaptive Jordan-Wigner transformation, that avoids nonlocal string operators. Primitives of such a reordering scheme are highlighted. Fermionic unitary circuits can be contracted with the same complexity as in the spin case. The scheme gives rise to a variational description of fermionic models not suffering from a sign problem. We present numerical examples on $9times 9$ and $6times 6$ fermionic lattice model to show the functioning of the approach.
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that are capable of implementing a universal set of gates with error rates below approximately 1%, requirements compatible with experimental reality. Consequently, a number of authors are attempting to squeeze additional performance out of the surface code. We describe an optimal complexity error suppression algorithm, parallelizable to O(1) given constant computing resources per unit area, and provide evidence that this algorithm exploits correlations in the error models of each gate in an asymptotically optimal manner.
Let $H_1, H_2$ be Hilbert spaces of the same finite dimension $ge2$, and $C$ an arbitrary quantum circuit with (principal) input state in $H_1$ and (principal) output state in $H_2$. $C$ may use ancillas and produce garbage which is traced out. $C$ may employ classical channels and measurement gates. If $C$ computes, for each computation path $mu$ through the circuit, a unitary transformation $U_mu: H_1 to H_2$ then, for each $mu$, the probability that a computation takes path $mu$ is independent of the input.
Dual-unitary quantum circuits can be used to construct 1+1 dimensional lattice models for which dynamical correlations of local observables can be explicitly calculated. We show how to analytically construct classes of dual-unitary circuits with any desired level of (non-)ergodicity for any dimension of the local Hilbert space, and present analytical results for thermalization to an infinite-temperature Gibbs state (ergodic) and a generalized Gibbs ensemble (non-ergodic). It is shown how a tunable ergodicity-inducing perturbation can be added to a non-ergodic circuit without breaking dual-unitarity, leading to the appearance of prethermalization plateaux for local observables.
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