In reference Bravyi et al., [Nature (London) 627, 778 (2024)NATUAS0028-083610.1038/s41586-024-07107-7], Bravyi et al. found examples of bivariate bicycle (BB) codes with similar logical performance to the surface code but with an improved encoding rate. In this work, we generalize a novel parity-check circuit design principle called morphing circuits and apply it to BB codes. We define a new family of BB codes whose parity check circuits require a qubit connectivity of degree five instead of six while maintaining their numerical performance. Logical input or output to an ancillary surface code is also possible in a biplanar layout. Finally, we develop a general framework for designing morphing circuits and present a sufficient condition for its applicability to two-block group algebra codes.

Neural network decoders can achieve a lower logical error rate compared to conventional decoders, like minimum-weight perfect matching, when decoding the surface code. Furthermore, these decoders require no prior information about the physical error rates, making them highly adaptable. In this study, we investigate the performance of such a decoder using both simulated and experimental data obtained from a transmon-qubit processor, focusing on small-distance surface codes. We first show that the neural network typically outperforms the matching decoder due to better handling of errors leading to multiple correlated syndrome defects, such as Y errors. When applied to the experimental data of Google Quantum AI [R. Acharya, Nature (London) 614, 676 (2023)10.1038/s41586-022-05434-1], the neural network decoder achieves logical error rates approximately 25% lower than minimum-weight perfect matching, approaching the performance of a maximum-likelihood decoder. To demonstrate the flexibility of this decoder, we incorporate the soft information available in the analog readout of transmon qubits and evaluate the performance of this decoder in simulation using a symmetric Gaussian-noise model. Considering the soft information leads to an approximately 10% lower logical error rate, depending on the probability of a measurement error. The good logical performance, flexibility, and computational efficiency make neural network decoders well-suited for near-term demonstrations of quantum memories.

Simulating noninteracting fermion systems is a common task in computational many-body physics. In the absence of translational symmetries, modeling free fermions on N modes usually requires poly(N) computational resources. While often moderate, these costs can be prohibitive in practice when large systems are considered. We present several free fermion problems that can be solved by a quantum algorithm with substantially reduced computational costs. The memory costs are exponentially improved, polylog(N). The run-time improvement, compared to the best known classical algorithms, is either exponential or significantly polynomial, depending on the geometry of the problem. The simulation of free fermion dynamics belongs to the BQP-hard complexity class (i.e., as hard as any (decision) problem that can be solved on a quantum computer). This implies (under standard assumptions) that our algorithm yields an exponential speedup for any classical algorithm at least for some geometries. The key technique in our algorithm is the block encoding of objects such as correlation matrices and Green's functions into a unitary. We demonstrate how such unitaries can be efficiently realized as quantum circuits, in the context of dynamics and thermal states of tight-binding Hamiltonians. The special cases of disordered and inhomogeneous lattices, as well as large nonlattice graphs, are presented in detail. Finally, we show that our simulation algorithm generalizes to other promising targets, including free-boson systems.

We introduce the fermionic satisfiability problem, Fermionic k-SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on n fermionic modes, where each fermionic projector involves at most k fermionic modes. We prove that this problem can be solved efficiently classically for k = 2. In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA₁-hard.

One of the main problems in computational physics is predicting the low-energy behavior of many-body quantum systems. The computational complexity of this problem, however, is relatively poorly understood. A recent major progress in this direction has been the no low-energy trivial states (NLTS) theorem; it gives a family of qubit Hamiltonians whose low-energy states cannot be reached by shallow quantum circuits. In this work we provide a fermionic counterpart to this theorem, constructing local fermionic Hamiltonians with no low-energy trivial states. Distinct from the qubit case, we define trivial states via finite-depth fermionic quantum circuits. We further strengthen the result, allowing free access to (generally, deep) Gaussian fermionic circuits into our notion of a trivial state. The desired fermionic Hamiltonian can be constructed using any qubit Hamiltonian which has the NLTS property via well-spread distributions over bitstrings. We also define a fermionic analog of quantum probabilistically checkable proofs (PCPs) and explore the relation of fermionic PCP class with the qubit version.

Quantum error correction allows for quantum information to be preserved using logical qubits, which are subject to lower error rates than their constituent physical qubits. The degree of error suppression depends on the choice of error correcting code and distance, the underlying physical error rate, and the accuracy of the decoder. While traditional decoders utilise a binary (hard) syndrome, recent work shows that additional (soft) information captured during qubit readout can be effectively utilised to improve decoding accuracy. In this work, we present experimental results from a distance-three surface code implemented on transmon qubits, where we perform Z-stabiliser measurements to protect the state of the logical qubit against bit-flip errors. We initialise the logical qubit in one of 16 possible computational states representing the logical zero state, and perform repeated stabiliser checks over a variable number of rounds to preserve the state over time. We compare the decoding performance for a hard minimum-weight perfect matching decoder against a soft variant where rich measurement information is incorporated, and demonstrate an improved logical fidelity. Additionally, we employ a recurrent neural network decoder with both soft and hard variants and observe improved performance when soft information is used. The general nature of soft information makes it widely applicable to different physical qubit platforms, where it can be leveraged to shorten measurement times and improve the logical fidelity in quantum error correction experiments. Pre-print available at arXiv:2403.00706.

We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense case [1, 2], we prove that strictly q-local sparse fermionic Hamiltonians have a constant Gaussian approximation ratio; the result holds for any connectivity and interaction strengths. Sparsity means that each fermion participates in a bounded number of interactions, and strictly q-local means that each term involves exactly q fermionic (Majorana) operators. We extend our proof to give a constant Gaussian approximation ratio for sparse fermionic Hamiltonians with both quartic and quadratic terms. With additional work, we also prove a constant Gaussian approximation ratio for the so-called sparse SYK model with strictly 4-local interactions (sparse SYK-4 model). In each setting we show that the Gaussian state can be efficiently determined. Finally, we prove that the O(n⁻¹/²) Gaussian approximation ratio for the normal (dense) SYK-4 model extends to SYK-q for even q > 4, with an approximation ratio of O(n¹/^2−q/4). Our results identify non-sparseness as the prime reason that the SYK-4 model can fail to have a constant approximation ratio [1, 2].

Minimizing leakage from computational states is a challenge when using many-level systems like superconducting quantum circuits as qubits. We realize and extend the quantum-hardware-efficient, all-microwave leakage reduction unit (LRU) for transmons in a circuit QED architecture proposed by Battistel et al. This LRU effectively reduces leakage in the second- and third-excited transmon states with up to 99% efficacy in 220 ns, with minimum impact on the qubit subspace. As a first application in the context of quantum error correction, we show how multiple simultaneous LRUs can reduce the error detection rate and suppress leakage buildup within 1% in data and ancilla qubits over 50 cycles of a weight-2 stabilizer measurement.

Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, also when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a ‘phase function’ g(k) = ^P_j A_je^iφjk with unknown eigenphases φ_j and overlaps A_j at quantum cost O(k), we show how to estimate the phases {φ_j} with (root-mean-square) error δ for total quantum cost T = O(δ⁻¹). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [1, 2] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [3] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.

Solid-state spin qubits is a promising platform for quantum computation and quantum networks^1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems^3–8, elementary quantum algorithms^8–11 and non-fault-tolerant error correction^12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations^15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science^15,19–27. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits^28–30. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.

We consider the task of spectral estimation of local quantum Hamiltonians. The spectral estimation is performed by estimating the oscillation frequencies or decay rates of signals representing the time evolution of states. We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation (QPE) circuits, assuming efficient preparation of the input state. We prove that our MC scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of MC methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the quantum-phase-estimation-based eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain.

We propose and analyze two types of microwave-activated gates between a fluxonium and a transmon qubit, namely a cross-resonance (CR) and a CPHASE gate. The large frequency difference between a transmon and a fluxonium makes the realization of a two-qubit gate challenging. For a medium-frequency fluxonium qubit, the transmon-fluxonium system allows for a cross-resonance effect mediated by the higher levels of the fluxonium over a wide range of transmon frequencies. This allows one to realize the cross-resonance gate by driving the fluxonium at the transmon frequency, mitigating typical problems of the cross-resonance gate in transmon-transmon chips related to frequency targeting and residual ZZ coupling. However, when the fundamental frequency of the fluxonium enters the low-frequency regime below 100MHz, the cross-resonance effect decreases leading to long gate times. For this range of parameters, a fast microwave CPHASE gate can be implemented using the higher levels of the fluxonium. In both cases, we perform numerical simulations of the gate showing that a gate fidelity above 99% can be obtained with gate times between 100 and 300ns. Next to a detailed gate analysis, we perform a study of chip yield for a surface code lattice of fluxonia and transmons interacting via the proposed cross-resonance gate. We find a much better yield as compared to a transmon-only architecture with the cross-resonance gate as native two-qubit gate.

The fault-tolerant operation of logical qubits is an important requirement for realizing a universal quantum computer. Spin qubits based on quantum dots have great potential to be scaled to large numbers because of their compatibility with standard semiconductor manufacturing. Here, we show that a quantum error correction code can be implemented using a four-qubit array in germanium. We demonstrate a resonant SWAP gate and by combining controlled-Z and controlled-S⁻¹ gates we construct a Toffoli-like three-qubit gate. We execute a two-qubit phase flip code and find that we can preserve the state of the data qubit by applying a refocusing pulse to the ancilla qubit. In addition, we implement a phase flip code on three qubits, making use of a Toffoli-like gate for the final correction step. Both the quality and quantity of the qubits will require significant improvement to achieve fault-tolerance. However, the capability to implement quantum error correction codes enables co-design development of quantum hardware and software, where codes tailored to the properties of spin qubits and advances in fabrication and operation can now come together to advance semiconductor quantum technology.

We analyze whether circuit QED Hamiltonians are stoquastic, focusing on systems of coupled flux qubits. We show that scalable sign-problem-free path integral Monte Carlo simulations can typically be performed for such systems. Despite this, we corroborate the recent finding [I. Ozfidan, Phys. Rev. Appl. 13, 034037 (2020)10.1103/PhysRevApplied.13.034037] that an effective, nonstoquastic qubit Hamiltonian can emerge in a system of capacitively coupled flux qubits. We find that if the capacitive coupling is sufficiently small, this nonstoquasticity of the effective qubit Hamiltonian can be avoided if we perform a canonical transformation prior to projecting onto an effective qubit Hamiltonian. Our results shed light on the power of circuit QED Hamiltonians for the use of quantum adiabatic computation and the subtlety of finding a representation which cures the sign problem in these systems.

Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC). We propose a scheme using two leakage-reduction units (LRUs) that mitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QEC-cycle time as in previous proposals. For data qubits, we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly disturbs the computational states for realistic system parameters. For ancilla qubits, we apply a |1↔|2π pulse conditioned on the measurement outcome. Using density-matrix simulations of the distance-3 surface code, we show that the average leakage lifetime is reduced to almost one QEC cycle, even when the LRUs are implemented with limited fidelity. Furthermore, we show that this leads to a significant reduction of the logical error rate. This LRU scheme opens the prospect for near-term scalable QEC demonstrations.

We present a Dicke state preparation scheme which uses global control of N spin qubits: our scheme is based on the standard phase estimation algorithm, which estimates the eigenvalue of a unitary operator. The scheme prepares a Dicke state nondeterministically by collectively coupling the spins to an ancilla qubit via a ZZ interaction, using log2N+1 ancilla qubit measurements. The preparation of such Dicke states can be useful if the spins in the ensemble are used for magnetic sensing: we discuss a possible realization using an ensemble of electronic spins located at diamond nitrogen-vacancy centers coupled to a single superconducting flux qubit. We also analyze the effect of noise and limitations in our scheme.