State-of-the-art classical neural networks are observed to be vulnerable to small crafted adversarial perturbations. A more severe vulnerability has been noted for quantum machine learning (QML) models classifying Haar-random pure states. This stems from the concentration of measure phenomenon, a property of the metric space when sampled probabilistically, and is independent of the classification protocol. In order to provide insights into the adversarial robustness of a quantum classifier on real-world classification tasks, we focus on the adversarial robustness in classifying a subset of encoded states that are smoothly generated from a Gaussian latent space. We show that the vulnerability of this task is considerably weaker than that of classifying Haar-random pure states. In particular, we find only mildly polynomially decreasing robustness in the number of qubits, in contrast to the exponentially decreasing robustness when classifying Haar-random pure states and suggesting that QML models can be useful for real-world classification tasks.

State-of-the-art classical neural networks are observed to be vulnerable to small crafted adversarial perturbations. A more severe vulnerability has been noted for quantum machine learning (QML) models classifying Haar-random pure states. This stems from the concentration of measure phenomenon, a property of the metric space when sampled probabilistically, and is independent of the classification protocol. To provide insights into the adversarial robustness of a quantum classifier on real-world classification tasks, we focus on the adversarial robustness in classifying a subset of encoded states that are smoothly generated from a Gaussian latent space. We show that the vulnerability of this task is considerably weaker than that of classifying Haar-random pure states. In particular, we find only mildly polynomially decreasing robustness in the number of qubits, in contrast to the exponentially decreasing robustness when classifying Haar-random pure states and suggesting that QML models can be useful for real-world classification tasks.

Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).

We introduce a unitary coupled-cluster (UCC) ansatz termed k-UpCCGSD that is based on a family of sparse generalized doubles operators, which provides an affordable and systematically improvable unitary coupled-cluster wave function suitable for implementation on a near-term quantum computer. k-UpCCGSD employs k products of the exponential of pair coupled-cluster double excitation operators (pCCD), together with generalized single excitation operators. We compare its performance in both efficiency of implementation and accuracy with that of the generalized UCC ansatz employing the full generalized single and double excitation operators (UCCGSD), as well as with the standard ansatz employing only single and double excitations (UCCSD). k-UpCCGSD is found to show the best scaling for quantum computing applications, requiring a circuit depth of [Formula: see text], compared with [Formula: see text] for UCCGSD, and [Formula: see text] for UCCSD, where N is the number of spin orbitals and η is the number of electrons. We analyzed the accuracy of these three ansätze by making classical benchmark calculations on the ground state and the first excited state of H₄ (STO-3G, 6-31G), H₂O (STO-3G), and N₂ (STO-3G), making additional comparisons to conventional coupled cluster methods. The results for ground states show that k-UpCCGSD offers a good trade-off between accuracy and cost, achieving chemical accuracy for lower cost of implementation on quantum computers than both UCCGSD and UCCSD. UCCGSD is also found to be more accurate than UCCSD but at a greater cost for implementation. Excited states are calculated with an orthogonally constrained variational quantum eigensolver approach. This is seen to generally yield less accurate energies than for the corresponding ground states. We demonstrate that using a specialized multideterminantal reference state constructed from classical linear response calculations allows these excited state energetics to be improved.

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states¹. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases^2-8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)^7,9-15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order^7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states^7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors.

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 correction^15-17 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 device^18,19 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.