High-fidelity spin qubit operation and algorithmic initialization above 1 K.

The encoding of qubits in semiconductor spin carriers has been recognized as a promising approach to a commercial quantum computer that can be lithographically produced and integrated at scale1-10. However, the operation of the large number of qubits required for advantageous quantum applications11-13 will produce a thermal load exceeding the available cooling power of cryostats at millikelvin temperatures. As the scale-up accelerates, it becomes imperative to establish fault-tolerant operation above 1 K, at which the cooling power is orders of magnitude higher14-18. Here we tune up and operate spin qubits in silicon above 1 K, with fidelities in the range required for fault-tolerant operations at these temperatures19-21. We design an algorithmic initialization protocol to prepare a pure two-qubit state even when the thermal energy is substantially above the qubit energies and incorporate radiofrequency readout to achieve fidelities up to 99.34% for both readout and initialization. We also demonstrate single-qubit Clifford gate fidelities up to 99.85% and a two-qubit gate fidelity of 98.92%. These advances overcome the fundamental limitation that the thermal energy must be well below the qubit energies for the high-fidelity operation to be possible, surmounting a main obstacle in the pathway to scalable and fault-tolerant quantum computation.

Fault-Tolerant Thresholds for the Surface Code in Excess of 5% under Biased Noise.

Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalizes to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼5% in an experimentally relevant regime where dephasing errors occur at a rate 100 times greater than bit-flip errors.

Ultrahigh Error Threshold for Surface Codes with Biased Noise.

We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli Z errors occur more frequently than X or Y errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara, and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is, in fact, at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of Y instead of Z around the faces, as this doubles the number of useful syndrome bits associated with the dominant Z errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes.

Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities.

We present a method for estimating the probabilities of outcomes of a quantum circuit using Monte Carlo sampling techniques applied to a quasiprobability representation. Our estimate converges to the true quantum probability at a rate determined by the total negativity in the circuit, using a measure of negativity based on the 1-norm of the quasiprobability. If the negativity grows at most polynomially in the size of the circuit, our estimator converges efficiently. These results highlight the role of negativity as a measure of nonclassical resources in quantum computation.

Topological entanglement entropy with a twist.

Defects in topologically ordered models have interesting properties that are reminiscent of the anyonic excitations of the models themselves. For example, dislocations in the toric code model are known as twists and possess properties that are analogous to Ising anyons. We strengthen this analogy by using the topological entanglement entropy as a diagnostic tool to identify properties of both defects and excitations in the toric code. Specifically, we show, through explicit calculation, that the toric code model including twists and dyon excitations has the same quantum dimensions, the same total quantum dimension, and the same fusion rules as an Ising anyon model.

Symmetry-protected phases for measurement-based quantum computation.

Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.

Low-overhead fault-tolerant quantum computing using long-range connectivity.

Vast numbers of qubits will be needed for large-scale quantum computing because of the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes and the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around 100 logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.

Programming a quantum phase of matter.

The ability to measure long-range entanglement may enable robust quantum memory.

The XZZX surface code.

Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code-the XZZX code-offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

Nonclassical correlations from randomly chosen local measurements.

We show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states. Specifically, spatially separated parties who perform local measurements on a maximally entangled state using randomly chosen measurement bases can, with significant probability, generate nonclassical correlations that violate a Bell inequality. For n parties using a Greenberger-Horne-Zeilinger state, this probability of violation rapidly tends to unity as the number of parties increases. We also show that, even with both a randomly chosen two-qubit pure state and randomly chosen measurement bases, a violation can be found about 10% of the time. Among other applications, our work provides a feasible alternative for the demonstration of Bell inequality violation without a shared reference frame.

Quantum computational renormalization in the Haldane phase.

Single-spin measurements on the ground state of an interacting spin lattice can be used to perform a quantum computation. We show how such measurements can mimic renormalization group transformations and remove the short-ranged variations of the state that can reduce the fidelity of a computation. This suggests that the quantum computational ability of a spin lattice could be a robust property of a quantum phase. We illustrate our idea with the ground state of a rotationally invariant spin-1 chain, which can serve as a quantum computational wire not only at the Affleck-Kennedy-Lieb-Tasaki point, but within the Haldane phase.

Fast spin exchange across a multielectron mediator.

Scalable quantum processors require tunable two-qubit gates that are fast, coherent and long-range. The Heisenberg exchange interaction offers fast and coherent couplings for spin qubits, but is intrinsically short-ranged. Here, we demonstrate that its range can be increased by employing a multielectron quantum dot as a mediator, while preserving speed and coherence of the resulting spin-spin coupling. We do this by placing a large quantum dot with 50-100 electrons between a pair of two-electron double quantum dots that can be operated and measured simultaneously. Two-spin correlations identify coherent spin-exchange processes across the multielectron quantum dot. We further show that different physical regimes of the mediated exchange interaction allow a reduced susceptibility to charge noise at sweet spots, as well as positive and negative coupling strengths up to several gigahertz. These properties make multielectron dots attractive as scalable, voltage-controlled coherent coupling elements.

Symmetry-respecting real-space renormalization for the quantum Ashkin-Teller model.

We use a simple real-space renormalization-group approach to investigate the critical behavior of the quantum Ashkin-Teller model, a one-dimensional quantum spin chain possessing a line of criticality along which critical exponents vary continuously. This approach, which is based on exploiting the on-site symmetry of the model, has been shown to be surprisingly accurate for predicting some aspects of the critical behavior of the quantum transverse-field Ising model. Our investigation explores this approach in more generality, in a model in which the critical behavior has a richer structure but which reduces to the simpler Ising case at a special point. We demonstrate that the correlation length critical exponent as predicted from this real-space renormalization-group approach is in broad agreement with the corresponding results from conformal field theory along the line of criticality. Near the Ising special point, the error in the estimated critical exponent from this simple method is comparable to that of numerically intensive simulations based on much more sophisticated methods, although the accuracy decreases away from the decoupled Ising model point.

Efficient quantum state tomography.

Quantum state tomography--deducing quantum states from measured data--is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. One of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

Identifying phases of quantum many-body systems that are universal for quantum computation.

Quantum computation can proceed solely through single-qubit measurements on an appropriate quantum state, such as the ground state of an interacting many-body system. We investigate a simple spin-lattice system based on the cluster-state model, and by using nonlocal correlation functions that quantify the fidelity of quantum gates performed between distant qubits, we demonstrate that it possesses a quantum (zero-temperature) phase transition between a disordered phase and an ordered "cluster phase" in which it is possible to perform a universal set of quantum gates.

Demonstrating superior discrimination of locally prepared states using nonlocal measurements.

We experimentally demonstrate the superior discrimination of separated, unentangled two-qubit correlated states using nonlocal measurements, when compared with measurements based on local operations and classical communications. When predicted theoretically, this phenomenon was dubbed "quantum nonlocality without entanglement." We characterize the performance of the nonlocal, or joint, measurement with a payoff function, for which we measure 0.72 +/- 0.02, compared with the maximum locally achievable value of 2/3 and the overall optimal value of 0.75.