Fault-tolerant Quantum Computation Research Articles

Early Fault-Tolerant Quantum Computing

In recent years, research in quantum computing has largely focused on two approaches: near-term intermediate-scale quantum (NISQ) computing and future fault-tolerant quantum computing (FTQC). A growing body of research into early fault-tolerant quantum computing (EFTQC) is exploring how to utilize quantum computers during the transition between these two eras. However, without agreed-upon characterizations of this transition, it is unclear how best to utilize EFTQC architectures. We argue for the perspective that this transition period will be characterized by a law of diminishing returns in quantum error correction (QEC), where the ability of the architecture to maintain quality operations at scale determines the point of diminishing returns. Two challenges emerge from this picture: how to model this phenomenon of diminishing return of QEC as the performance of devices is continually improving and how to design algorithms to make the most use of these devices. To address these challenges, we present models for the performance of EFTQC architectures, capturing the diminishing returns of QEC. We then use these models to elucidate the regimes in which algorithms suited to such architectures are advantageous. As a concrete example, we show that for the canonical task of phase estimation, in a regime of moderate scalability and using just over one million physical qubits, the “reach” of the quantum computer can be extended (compared to the standard approach) from 90-qubit instances to over 130-qubit instances using a simple early fault-tolerant quantum algorithm, which reduces the number of operations per circuit by a factor of 100 and increases the number of circuit repetitions by a factor of 10 000. This clarifies the role that such algorithms might play in the era of limited-scalability quantum computing. Published by the American Physical Society 2024

Beyond the standard model of topological Josephson junctions: From crystalline anisotropy to finite-size and diode effects

A planar Josephson junction is a versatile platform to realize topological superconductivity over a large parameter space and host Majorana bound states. With a change in the Zeeman field, this system undergoes a transition from trivial to topological superconductivity accompanied by a jump in the superconducting phase difference between the two superconductors. A standard model of these Josephson junctions, which can be fabricated to have a nearly perfect interfacial transparency, predicts a simple universal behavior. In that model, at the same value of Zeeman field for the topological transition, there is a π phase jump and a minimum in the critical superconducting current, while applying a controllable phase difference yields a diamond-shaped topological region as a function of that phase difference and a Zeeman field. In contrast, even for a perfect interfacial transparency, we find a much richer and nonuniversal behavior as the width of the superconductor is varied or the Dresselhaus spin–orbit coupling is considered. The Zeeman field for the phase jump, not necessarily π, is different from the value for the minimum critical current, while there is a strong deviation from the diamond-like topological region. These Josephson junctions show a striking example of a nonreciprocal transport and superconducting diode effect, revealing the importance of our findings not only for topological superconductivity and fault-tolerant quantum computing but also for superconducting spintronics.

Deterministic Preparation of Optical Squeezed Cat and Gottesman-Kitaev-Preskill States.

Large-amplitude squeezed cat states and high-quality Gottesman-Kitaev-Preskill (GKP) states are essential for effective quantum error correction, yet their optical preparation has been hindered by challenges such as low success probabilities, small amplitudes, and insufficient squeezing. Addressing these limitations, our research introduces scalable optical schemes for the deterministic preparation of large-amplitude squeezed cat states from photon-number states. Fock states have the benefit of producing consistent cat states across all measurement outcomes and intrinsically provides a degree of squeezing. Notably, these squeezed cat states facilitate the deterministic generation of high-quality approximate GKP states via "breeding," showing that GKP error correction in optics is technically feasible in near-term experiments. Our schemes allow fault-tolerant quantum computation through the use of GKP error correction.

Critical quantum geometric tensors of parametrically-driven nonlinear resonators

Parametrically driven nonlinear resonators represent a building block for realizing fault-tolerant quantum computation and are useful for critical quantum sensing. From a fundamental viewpoint, the most intriguing feature of such a system is perhaps the critical phenomena, which can occur without interaction with any other quantum system. The non-analytic behaviors of its eigenspectrum have been substantially investigated, but those associated with the ground state wavefunction have largely remained unexplored. Using the quantum ground state geometric tensor as an indicator, we comprehensively establish a phase diagram involving the driving parameter ε and phase ϕ. The results reveal that with the increase in ε, the system undergoes a quantum phase transition from the normal to the symmetry-breaking phase, with the critical point unaffected by ϕ. Furthermore, the critical exponent and scaling dimension are obtained by an exact numerical method, which is consistent with previous works. Our numerical results show that the phase transition falls within the universality class of the quantum Rabi model. This work reveals that the quantum metric and Berry curvature display diverging behaviors across the quantum phase transition.

High-efficiency realization of the super-robust Rydberg Deutsch gate

Quantum logic gates are the essential components of quantum computation and have broad practical applications, especially for the multiple-qubit logic gates. Compared with appliying a series of single- and two-qubit gates, constrcuting quantum computation with multiple-qubit logic gates can be more efficient and high-fidelity. As a three-qubit logic gate, Deutsch gate enables the realization of any feasible quantum computations. Based on neutral atoms, we present a scheme of super-robust Deutsch gate via optimal control technique. Utilizing the Rydberg blockade effect of neutral atoms, we design an implementable D(β) based on the three-step program. One of the notable advantages of this function is that β can be achieved arbitrarily between 0 and π with the operation of target atom. In addition, we give an analytical solution agreed very well with numerical simulations for the residual blocking effect between next-neighbor atoms that affects the performance quality of Deutsch gate. The fidelity of our proposed scheme is demonstrated by numerical simulation of the master equation based on the full Hamiltonian, and the robustness of our scheme with control errors is also illustrated. Our proposal offers an alternative promising scheme for fault-tolerant quantum computation. Published by the American Physical Society 2024

Factorization of large tetra and penta prime numbers on IBM quantum processor

The factorization of large digit integers in polynomial time is a challenging computational task to decipher. The development of Shor’s algorithm sparked a new resolution for solving the factorization problem. However, putting Shor’s algorithm into use in real-world situations presents major difficulties. The algorithm largely depends on the availability of large-scale, fault-tolerant quantum computers, which are not available at present. The need for qubit coherence and error correction makes the algorithm susceptible to noise and decoherence, hindering its practical realization. Therefore, exploring alternative quantum factorization algorithms and investing in quantum computing hardware advancements are vital steps toward overcoming these drawbacks and harnessing the full potential of quantum computing for factorization tasks. This article explores an alternative method of converting the factorization problem into an optimization problem using appropriate analytic algebra. The generalized Grover’s protocol is used to increase the amplitude of the necessary states and, in turn, help in the execution of the quantum factorization of tetra and penta primes as a proof of concept for different integers, including 875, 1 269 636 549 803, and 4375, using three and four qubits of IBMQ Perth (a seven-qubit processor). The fidelity of the quantum factorization protocol with the IBMQ Perth qubits was near unity. A generalization of the method is provided at the end for implementing factorization problems in various cases.

Quantum computation of stopping power for inertial fusion target design

Stopping power is the rate at which a material absorbs the kinetic energy of a charged particle passing through it-one of many properties needed over a wide range of thermodynamic conditions in modeling inertial fusion implosions. First-principles stopping calculations are classically challenging because they involve the dynamics of large electronic systems far from equilibrium, with accuracies that are particularly difficult to constrain and assess in the warm-dense conditions preceding ignition. Here, we describe a protocol for using a fault-tolerant quantum computer to calculate stopping power from a first-quantized representation of the electrons and projectile. Our approach builds upon the electronic structure block encodings of Su et al. [PRX Quant. 2, 040332 (2021)], adapting and optimizing those algorithms to estimate observables of interest from the non-Born-Oppenheimer dynamics of multiple particle species at finite temperature. We also work out the constant factors associated with an implementation of a high-order Trotter approach to simulating a grid representation of these systems. Ultimately, we report logical qubit requirements and leading-order Toffoli costs for computing the stopping power of various projectile/target combinations relevant to interpreting and designing inertial fusion experiments. We estimate that scientifically interesting and classically intractable stopping power calculations can be quantum simulated with roughly the same number of logical qubits and about one hundred times more Toffoli gates than is required for state-of-the-art quantum simulations of industrially relevant molecules such as FeMoco or P450.

Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes

Quantum error correction is crucial for scalable quantum information-processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present the first implementation of time-domain decoding with the overlapping window method for general QLDPC codes and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes. Published by the American Physical Society 2024

Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors

Topological color codes are widely acknowledged as promising candidates for fault-tolerant quantum computing. Neither a two-dimensional nor a three-dimensional topology, however, can provide a universal gate set {, , }, with the gate missing in the two-dimensional and the gate in the three-dimensional case. These complementary shortcomings of the isolated topologies may be overcome in a combined approach, by switching between a two- and a three-dimensional code while maintaining the logical state. In this work, we construct resource-optimized deterministic and nondeterministic code-switching protocols for two- and three-dimensional distance-three color codes using fault-tolerant quantum circuits based on flag qubits. Deterministic protocols allow for the fault-tolerant implementation of logical gates on an encoded quantum state, while nondeterministic protocols may be used for the fault-tolerant preparation of magic states. Taking the error rates of state-of-the-art trapped-ion quantum processors as a reference, we find a logical failure probability of 3% for deterministic logical gates, which cannot be realized transversally in the respective code. By replacing the three-dimensional distance-three color code in the protocol for magic state preparation with the morphed code introduced in Vasmer and Kubica [PRX Quantum 3, 030319 (2022)], we reduce the logical failure rates by 2 orders of magnitude, thus rendering it a viable method for magic state preparation on near-term quantum processors. Our results demonstrate that code switching enables the fault-tolerant and deterministic implementation of a universal gate set under realistic conditions, and thereby provide a practical avenue to advance universal, fault-tolerant quantum computing and enable quantum algorithms on first, error-corrected logical qubits. Published by the American Physical Society 2024

Reducing the Runtime of Fault-Tolerant Quantum Simulations in Chemistry through Symmetry-Compressed Double Factorization.

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

Catalyst-free MBE growth of PbSnTe nanowires with tunable aspect ratio

Topological crystalline insulators (TCIs) are interesting for their topological surface states, which hold great promise for scattering-free transport channels and fault-tolerant quantum computing. A promising TCI is SnTe. However, Sn-vacancies form in SnTe, causing a high hole density, hindering topological transport from the surface being measured. This issue could be relieved by using nanowires with a high surface-to-volume ratio. Furthermore, SnTe can be alloyed with Pb reducing the Sn-vacancies while maintaining its topological phase. Here we present the catalyst-free growth of monocrystalline PbSnTe in molecular beam epitaxy. By the addition of a pre-deposition stage before the growth, we have control over the nucleation phase and thereby increase the nanowire yield. This facilitates tuning the nanowire aspect ratio by a factor of four by varying the growth parameters. These results allow us to grow specific morphologies for future transport experiments to probe the topological surface states in a Pb1–x Sn x Te-based platform.

Nontrivial evolution and geometric phase for an orbital angular momentum qutrit

Photonic orbital angular momentum (OAM) offers a promising platform for high-dimensional quantum information processing. While geometric phase (GP) is the crucial tool in enabling intrinsically fault-tolerant quantum computation, the measurement of GP using linear optics remains relatively unexplored in the OAM state space. Here, we propose an experimental scheme to detect GP shifts resulting from the cyclic evolution of OAM qutrit states. Distinguished with the conventional evolution along cyclic path on the Poincaré sphere (PS), the nontrivial evolution in our theoretical scheme is along a cyclic path residing within the SU(3)/U(2) parameter space. By employing a combination of X-gates, dove prisms, and double cylindrical lenses, we achieve the cyclic evolution and analyse the resultant GP through our designed Sagnac interferometer. Our theoretical study may find potential in high-dimensional quantum computation using twisted photons and in exploring the geometric structure of such optical systems.

Pressure-Dependent Superconductivity in Topological Dirac Semimetal SrCuBi.

The discovery of superconducting states in diverse topological materials generates a burgeoning interest to explore a topological superconductor and to realize a fault-tolerant topological quantum computation. A variety of routes to realize topological superconductors are proposed, and many types of topological materials are developed. However, a pristine topological material with a natural superconducting state is relatively rare as compared to topological materials with artificially induced superconductivity. Here, it is reported that the planar honeycomb structured 3D topological Dirac semimetal (TDS) SrCuBi, which is the Zintl phase, shows a natural surface superconductivity at 2.1 K under ambient pressure. It is clearly identified from theoretical calculations that a topologically nontrivial state exists on the (100) surface. Further, its superconducting transition temperature (Tc) increases by applying pressure, exhibiting a maximal Tc of 4.8 K under 6.2GPa. It is believed that this discovery opens up a new possibility of exploring exotic Majorana fermions at the surface of 3D TDS superconductors.

Magic of quantum hypergraph states

Magic, or nonstabilizerness, characterizes the deviation of a quantum state from the set of stabilizer states, playing a fundamental role in quantum state complexity and universal fault-tolerant quantum computing. However, analytical and numerical characterizations of magic are very challenging, especially for multi-qubit systems, even with a moderate qubit number. Here, we systemically and analytically investigate the magic resource of archetypal multipartite quantum states—quantum hypergraph states, which can be generated by multi-qubit controlled-phase gates encoded by hypergraphs. We first derive the magic formula in terms of the stabilizer Rényi-α entropies for general quantum hypergraph states. If the average degree of the corresponding hypergraph is constant, we show that magic cannot reach the maximal value, i.e., the number of qubits n. Then, we investigate the statistical behaviors of random hypergraph states' magic and prove a concentration result, indicating that random hypergraph states typically reach the maximum magic. This also suggests an efficient way to generate maximal magic states with random diagonal circuits. Finally, we study hypergraph states with permutation symmetry, such as 3-complete hypergraph states, where any three vertices are connected by a hyperedge. Counterintuitively, such states can only possess constant or even exponentially small magic for α≥2. Our study advances the understanding of multipartite quantum magic and could lead to applications in quantum computing and quantum many-body physics.

Gate-Controlled Anyon Generation and Detection in Kitaev Spin Liquids.

Reliable manipulation of non-Abelian Ising anyons supported by Kitaev spin liquids may enable intrinsically fault-tolerant quantum computation. Here, we introduce a standalone scheme for both generating and detecting individual Ising anyons using tunable gate voltages in a heterostructure containing a non-Abelian Kitaev spin liquid and a monolayer semiconductor. The key ingredients of our setup are a Kondo coupling to stabilize an Ising anyon in the spin liquid around each electron in the semiconductor, and a large charging energy to allow control over the electron numbers in distinct gate-defined regions of the semiconductor. In particular, a single Ising anyon can be generated at a disk-shaped region by gate tuning its electron number to one, while it can be interferometrically detected by measuring the electrical conductance of a ring-shaped region around it whose electron number is allowed to fluctuate between zero and one. We provide concrete experimental guidelines for implementing our proposal in promising candidate materials like α-RuCl_{3}.

Fault-tolerant quantum computation using low-cost joint measurements

We introduce a new method to implement joint measurements using a 4-qubit twist defect on a rotated surface code. The proposed method enables us to perform logical S (Phase gate), T (π/8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi /8$$\\end{document} gate), and H (Hadamard) with low overhead. Combined with other universal quantum gates, we can implement fault-tolerant quantum computation at the lattice surgery level beyond the gate level while saving considerable resources. We compare our method with previous methods using benchmark circuits by calculating the space and time costs. The proposed method requires additional lines of physical qubits for each encoded patch. Although it slightly increases the space cost for logical H compared to the previous work, it reduces the time cost. In addition, the proposed method decreases the space cost and time cost by introducing a 4-qubit twist defect for logical S and T. Therefore, the overall space-time cost is reduced.

Partial Syndrome Measurement for Hypergraph Product Codes

Hypergraph product codes are a promising avenue to achieving fault-tolerant quantum computation with constant overhead. When embedding these and other constant-rate qLDPC codes into 2D, a significant number of nonlocal connections are required, posing difficulties for some quantum computing architectures. In this work, we introduce a fault-tolerance scheme that aims to alleviate the effects of implementing this nonlocality by measuring generators acting on spatially distant qubits less frequently than those which do not. We investigate the performance of a simplified version of this scheme, where the measured generators are randomly selected. When applied to hypergraph product codes and a modified small-set-flip decoding algorithm, we prove that for a sufficiently high percentage of generators being measured, a threshold still exists. We also find numerical evidence that the logical error rate is exponentially suppressed even when a large constant fraction of generators are not measured.

Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 ms

Superconducting qubits are among the most advanced candidates for achieving fault-tolerant quantum computing. Despite recent significant advancements in the qubit lifetimes, the origin of the loss mechanism for state-of-the-art qubits is still subject to investigation. Furthermore, the successful implementation of quantum error correction requires negligible correlated errors between qubits. Here, we realize long-lived superconducting transmon qubits that exhibit fluctuating lifetimes, averaging 0.2 ms and exceeding 0.4 ms – corresponding to quality factors above 5 million and 10 million, respectively. We then investigate their dominant error mechanism. By introducing novel time-resolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bit-flip errors. Our findings not only deepen our understanding of the qubit error mechanisms but also provide valuable insights into potential error-mitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments.

Experimental Direct Quantum Fidelity Learning via a Data-Driven Approach.

Fidelity estimation is an important technique for evaluating prepared quantum states in noisy quantum devices. A recent theoretical work proposed a frugal approach called neural quantum fidelity estimation (NQFE) [X. Zhang et al., Phys. Rev. Lett. 127, 130503 (2021).PRLTAO0031-900710.1103/PhysRevLett.127.130503]. While this requires a much smaller number of measurement operators than full quantum state tomography, it uses a weight-based floating measurement strategy that predetermines the top global Pauli operators that contribute the most to the fidelity and uses discrete fidelity intervals as predictions. In this Letter, we develop a measurement-fixed NQFE based on a transformer model which requires less measurement cost and can output continuous estimates of fidelity. Here we further experimentally apply the NQFE in a realistic situation using a nuclear spin quantum processor. We prepare the ground states of local Hamiltonians and arbitrary states and investigate how to estimate their fidelity with reference states, and we compare the fidelity estimation strategy with our and the original NQFE to conventional tomography. It is shown that NQFE can estimate the fidelity with comparable accuracy to the tomography approach. In the future, NQFE will become an important tool for benchmarking quantum states ahead of the advent of well-trusted fault-tolerant quantum computers.

Reduced-order modeling on a near-term quantum computer

Quantum computing is an advancing area of research in which computer hardware and algorithms are developed to take advantage of quantum mechanical phenomena. In recent studies, quantum algorithms have shown promise in solving linear systems of equations as well as systems of linear ordinary differential equations (ODEs) and partial differential equations (PDEs). Reduced-order modeling (ROM) algorithms for studying fluid dynamics have shown success in identifying linear operators that can describe flowfields, where dynamic mode decomposition (DMD) is a particularly useful method in which a linear operator is identified from data. In this work, DMD is reformulated as an optimization problem to propagate the state of the linearized dynamical system on a quantum computer. This reformulation was chosen as a means of facilitating implementation on a near-term quantum computer. Quadratic unconstrained binary optimization (QUBO), a technique for optimizing quadratic polynomials in binary variables, allows for quantum annealing algorithms to be applied. A quantum circuit model (quantum approximation optimization algorithm, QAOA) is utilized to obtain predictions of the state trajectories. Results are shown for the quantum-ROM predictions for flow over a 2D cylinder at Re = 220 and flow over a NACA0009 airfoil at Re = 500 and α=15∘. The quantum-ROM predictions are found to depend on the number of bits utilized for a fixed point representation and the truncation level of the DMD model. Comparisons with DMD predictions from a classical computer algorithm are made, as well as an analysis of the computational complexity and prospects for future, more fault-tolerant quantum computers.