We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O(λ/ ) where λ is an absolute sum of Hamiltonian coefficients and is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity O(N + log(1/ )) where N is number of orbitals in the basis. This enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 / + N 2 log(1/ )/ ). Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code faulttolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.

In the past years, quantum computers more and more have evolved from an academic idea to an upcoming reality. IBM's project IBM Q can be seen as evidence of this progress. Launched in March 2017 with the goal to provide access to quantum computers for a broad audience, this allowed users to conduct quantum experiments on a 5-qubit and, since June 2017, also on a 16-qubit quantum computer (called IBM QX2 and IBM QX3, respectively). Revised versions of these 5-qubit and 16-qubit quantum computers (named IBM QX4 and IBM QX5, respectively) are available since September 2017. In order to use these, the desired quantum functionality (e.g. provided in terms of a quantum circuit) has to be properly mapped so that the underlying physical constraints are satisfied -a complex task. This demands solutions to automatically and efficiently conduct this mapping process.In this paper, we propose a methodology which addresses this problem, i.e. maps the given quantum functionality to a realization which satisfies all constraints given by the architecture and, at the same time, keeps the overhead in terms of additionally required quantum gates minimal. The proposed methodology is generic, can easily be configured for similar future architectures, and is fully integrated into IBM's SDK. Experimental evaluations show that the proposed approach clearly outperforms IBM's own mapping solution. In fact, for many quantum circuits, the proposed approach determines a mapping to the IBM architecture within minutes, while IBM's solution suffers from long runtimes and runs into a timeout of 1 hour in several cases. As an additional benefit, the proposed approach yields mapped circuits with smaller costs (i.e. fewer additional gates are required). All implementations of the proposed methodology is publicly available at http://iic.

Realizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

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