The simulation of fermionic systems is among the most anticipated applications of quantum computing. We performed several quantum simulations of chemistry with up to one dozen qubits, including modeling the isomerization mechanism of diazene. We also demonstrated error-mitigation strategies based on N-representability that dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realized the Givens rotation approach to noninteracting fermion evolution, which we variationally optimized to prepare the Hartree-Fock wave function. This ubiquitous algorithmic primitive is classically tractable to simulate yet still generates highly entangled states over the computational basis, which allowed us to assess the performance of our hardware and establish a foundation for scaling up correlated quantum chemistry simulations.

We present two new constructions for the Toffoli gate which substantially reduce resource costs in fault-tolerant quantum computing. The first contribution is a Toffoli gate requiring Clifford operations plus only four T = exp(iπσ z /8) gates, whereas conventional circuits require seven T gates. An extension of this result is that adding n control inputs to a controlled gate requires 4n T gates, whereas the best prior result was 8n. The second contribution is a quantum circuit for the Toffoli gate which can detect a single σ z error occurring with probability p in any one of eight T gates required to produce the Toffoli. By post-selecting circuits that did not detect an error, the posterior error probability is suppressed to lowest order from 4p (or 7p, without the first contribution) to 28p 2 for this enhanced construction. In fault-tolerant quantum computing, this construction can reduce the overhead for producing logical Toffoli gates by an order of magnitude.

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We develop a procedure for distilling magic states used in universal quantum computing that requires substantially fewer initial resources than prior schemes. Our distillation circuit is based on a family of concatenated quantum codes that possess a transversal Hadamard operation, enabling each of these codes to distill the eigenstate of the Hadamard operator. A crucial result of this design is that low-fidelity magic states can be consumed to purify other high-fidelity magic states to even higher fidelity, which we call "multilevel distillation." When distilling in the asymptotic regime of infidelity $\epsilon \rightarrow 0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O(\epsilon^{2^r})$ approaches $2^r+1$, which comes close to saturating the conjectured bound in [Phys. Rev. A 86, 052329]. We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate $\epsilon_{\mathrm{in}} = 0.01$ to $\epsilon_{\mathrm{out}}$ in the range $10^{-5}$ to $10^{-40}$ is about $14\log_{10}(1/\epsilon_{\mathrm{out}}) - 40$; the efficiency of multilevel distillation dominates all other reported protocols when distilling Hadamard magic states from initial infidelity 0.01 to any final infidelity below $10^{-7}$. These methods are an important advance for magic-state distillation circuits in high-performance quantum computing, and they provide insight into the limitations of nearly resource-optimal quantum error correction.Comment: 10 pages, 4 figure

State distillation is the process of taking a number of imperfect copies of a particular quantum state and producing fewer better copies. Until recently, the lowest overhead method of distilling states produced a single improved |A〉 state given 15 input copies. New block code state distillation methods can produce k improved |A〉 states given 3k + 8 input copies, potentially significantly reducing the overhead associated with state distillation. We construct an explicit surface code implementation of block code state distillation and quantitatively compare the overhead of this approach to the old. We find that, using the best available techniques, for parameters of practical interest, block code state distillation does not always lead to lower overhead, and, when it does, the overhead reduction is typically less than a factor of three.

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.

The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of ln 2, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction.

Quantum computation requires qubits that satisfy often-conflicting criteria, including scalable control and long-lasting coherence [1]. One approach to creating a suitable qubit is to operate in an encoded subspace of several physical qubits. Though such encoded qubits may be particularly susceptible to leakage out of their computational subspace, they can be insensitive to certain noise processes [2, 3] and can also allow logical control with a single type of entangling interaction [4] while maintaining favorable features of the underlying physical system. Here we demonstrate a qubit encoded in a subsystem of three coupled electron spins confined in gated, isotopically enhanced silicon quantum dots [4, 5]. Using a modified "blind" randomized benchmarking protocol that determines both computational and leakage errors [6, 7], we show that unitary operations have an average total error of 0.35%, with 0.17% of that coming from leakage driven by interactions with substrate nuclear spins. This demonstration utilizes only the voltage-controlled exchange interaction for qubit manipulation and highlights the operational benefits of encoded subsystems, heralding the realization of high-quality encoded multi-qubit operations [4, 8].Electrons trapped in silicon heterostructures have many attractive features, including very long coherence times in isotopically enriched material [9, 10] and compatibility with standard fabrication techniques. Singlespin qubits have recently demonstrated high-fidelity RFcontrolled single-qubit operations [10, 11] and two-qubit gates using the exchange interaction [12][13][14]. However, using RF signals for single-qubit control requires a large, stable magnetic field and introduces challenges with crosstalk. Fortunately, electron spins are particularly well-suited to forming encoded qubits. Two coupled electron spins can be operated at near-zero magnetic field as a "singlet-triplet" qubit [15,16]. That qubit is insensitive to uniform magnetic field fluctuations but still requires a magnetic field gradient for universal control. Three coupled electrons [17] can form a qubit with a tunable electric dipole moment, which could enhance RF selectivity, or the exchange-only qubit, which can be universally controlled using only the exchange interaction and does not require synchronization of gate operations with a local oscillator. Exchange is highly local and can be accurately controlled with a large on-off ratio using only fast voltage pulses. The combination of these features makes the exchange-only qubit especially attractive b X2