Universal Fault-tolerant Quantum Research Articles

Implementing arbitrary multimode continuous-variable quantum gates with fixed non-Gaussian states and adaptive linear optics

Non-Gaussian quantum gates are essential components for optical quantum information processing. However, the efficient implementation of practically important multimode higher-order non-Gaussian gates has not been comprehensively studied. We propose a measurement-based method to directly implement general, multimode, and higher-order non-Gaussian gates using only fixed non-Gaussian ancillary states and adaptive linear optics. Compared to existing methods, our method allows for a more resource-efficient and experimentally feasible implementation of multimode gates that are important for various applications in optical quantum technology, such as the two-mode cubic quantum nondemolition gate or the three-mode continuous-variable Toffoli gate, and their higher-order extensions. Our results will expedite the progress toward fault-tolerant universal quantum computing with light. Published by the American Physical Society 2024

Efficient realization of quantum algorithms with qudits

The development of a universal fault-tolerant quantum computer that can solve efficiently various difficult computational problems is an outstanding challenge for science and technology. In this work, we propose a technique for an efficient implementation of quantum algorithms with multilevel quantum systems (qudits). Our method uses a transpilation of a circuit in the standard qubit form, which depends on the characteristics of a qudit-based processor, such as the number of available qudits and the number of accessible levels. This approach provides a qubit-to-qudit mapping and comparison to a standard realization of quantum algorithms highlighting potential advantages of qudits. We provide an explicit scheme of transpiling qubit circuits into sequences of single-qudit and two-qudit gates taken from a particular universal set. We then illustrate our method by considering an example of an efficient implementation of a 6-qubit quantum algorithm with qudits. In this particular example, we demonstrate how using qudits allows a decreasing amount of two-body interactions in the qubit circuit implementation. We expect that our findings are of relevance for ongoing experiments with noisy intermediate-scale quantum devices that operate with information carriers allowing qudit encodings, such as trapped ions and neutral atoms, as well as optical and solid-state systems.

Flying-cat parity checks for quantum error correction

Long range, multiqubit parity checks have applications in both quantum error correction and measurement-based entanglement generation. Such parity checks could be performed using qubit-state-dependent phase shifts on propagating pulses of light described by coherent states |α〉 of the electromagnetic field. We consider “flying-cat” parity checks based on an entangling operation that is quantum nondemolition for Schrödinger's cat states |α〉±|−α〉. This operation encodes parity information in the phase of maximally distinguishable coherent states |±α〉, which can be read out using a phase-sensitive measurement of the electromagnetic field. In contrast to many implementations, where single-qubit errors and measurement errors can be treated as independent, photon loss during flying-cat parity checks introduces errors on physical qubits at a rate that is anticorrelated with the probability for measurement errors. We analyze this trade-off for three-qubit parity checks, which are a requirement for universal fault-tolerant quantum computing with the subsystem surface code. We further show how a six-qubit entangled “tetrahedron” state can be prepared using these three-qubit parity checks. The tetrahedron state can be used as a resource for controlled quantum teleportation of a two-qubit state or as a source of shared randomness with potential applications in three-party quantum key distribution. Finally, we provide conditions for performing high-quality flying-cat parity checks in a state-of-the-art circuit QED architecture, accounting for qubit decoherence, internal cavity losses, and finite-duration pulses, in addition to transmission losses. Published by the American Physical Society 2024

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.

(Invited) Cryogenic Electronics for Quantum Computing ICs: What Can Bring FDSOI

To design efficient quantum computers, conventional electronics is required as close as possible to the quantum bit (qubit) devices, considering either superconducting or Si-spin qubits, for the read-out and control, thus reducing the need for wiring toward room temperature [1]. This need highlights the broad importance of exploring and developing low-temperature CMOS technologies, with operation temperatures ranging from 4.2K down to well below 1K. Moreover, the Si-spin qubit process is also compatible with the CMOS process, allowing both of them, in principle, to be monolithically integrated on a single chip [2], [3]. This could provide an essential building block for any large-scale quantum processors, by enabling the design of scalable close-to-qubit cryogenic electronics for massive qubit matrix indexation and, ultimately, development of fault-tolerant universal gate-based quantum computers [4].In this approach, FDSOI technologies appear as a valuable solution, with their unique property of being able to adjust its threshold voltage (Vth) as a function of the temperature by the mean of a back bias VB. FDSOI thus enables high performance and low-power electronics. However, there is currently no process design kit to design CMOS circuits at cryogenic temperature, whatever the technology node considered. Therefore, extensive electrical characterization is required, with the aim of building a future cryogenic compact model, or at least of providing analytical and physics based models to designers in order to help them to optimize their circuits [5].In this work, we present an overview of the performances of FDSOI CMOS technology down to deep cryogenic temperature, with specific attention to devices electrostatics, variability or thermal effects. Because these aspects are essential for the development of compact models and robust design tools, we present recent progresses in electrical characterization and analysis of FDSOI MOSFETs down to 100mK, and the underlying physics, highlighting the effect of the back bias VB.We show that transistors can fully operate down to 100mK, keeping the same efficiency of the back bias on the threshold voltage [6]. Overall performances are mostly improved at low temperature, with e.g. a high reduction of the subthreshold slope and of the off current, or a higher transconductance maximum value due to higher carrier mobility. The tunability of the threshold voltage on a wide range through the back-bias offers a great advantage to the FDSOI technology, allowing improved performances [7] and the optimization of circuit in cryogenic environment [8], [9]. Other important electrical characteristics have also been investigated down to very low temperature, such as variability and mismatch properties [7], [10], or low frequency noise [11]. As a result, we show that FDSOI technology can outperform other CMOS technologies in cryogenic operation.Thermal effects, which play a fundamental role in cryogenic electronics – operating at various temperature stages, with limited cooling power. The study of self-heating effects at cryogenic temperature in this regard provides valuable information on how to manage thermal effects, and is discussed [7], [15].Finally, modeling issues are also addressed. Besides serious numerical difficulties due to singularities appearing at low temperature when solving model equations, the change of important physical parameters (like carrier mobility, thermal conductivity,...) needs to be correctly taken into account in future compact or analytical models. In addition, new physical effects evidenced at low temperature also need to be understood, analytically modeled and included as well. In particular, transport exhibits evidence of strong intersubband scattering in the channel of thin film FDSOI transistors [12]. The equations describing the sub-threshold regime are also strongly modified [13], [14].[1] E. Charbon et al., IEDM, 2016.[2] P. Galy et al., IEEE J. Electron Devices Soc., 2018.[3] L. Le Guevel et al., Appl. Phys. Rev., 2020.[4] X. Xue et al., Nature, 2021.[5] C. Enz et al., EDM, 2020[6] B. Cardoso Paz et al., EUROSOI-ULIS, 2020.[7] M. Cassé et al., to be published in IEDM, 2022.[8] H. Bohuslavskyi et al., IEEE Trans. Electron Devices, 2018.[9] L. Le Guevel et al., ISSCC, 2020.[10] B. Cardoso Paz et al., VLSI, 2020.[11] B. Cardoso Paz et al., IEEE Trans. Electron Devices, 2020.[12] M. Cassé et al., Appl. Phys. Lett., 2020.[13] H. Bohuslavskyi et al., IEEE Electron Device Lett., 2019.[14] G. Ghibaudo et al., Solid. State. Electron., 2020.[15] K. Triantopoulos et al., IEEE Trans. Electron Devices, 2019.

Multiple silicon dangling-bond charge qubits for quantum computing: a Hilbert-space analysis of the Hamiltonian

Silicon-based dangling-bond charge qubit is one of the auspicious models for universal fault-tolerant solid-state quantum computing. In universal quantum computing, it is crucial to evaluate and characterize the computational Hilbert space and reduce the complexity and size of the computational space. Here, we recognize this problem to understand the complexity and characteristics of the Hilbert space in our dangling-bond qubit model. The size of the desired Hilbert space can prominently be reduced by considering assumptions regarding the qubit loss. Moreover, the dimension of the desired subsets in the space shrinks by a factor of two due to the spin preservation property. Finally, the required classical memory for storage of the qubit information, Hamiltonian and Hilbert space is analysed when the number of qubits grows.

Strategies for a practical advantage of fault-tolerant circuit design in noisy trapped-ion quantum computers

Fault-tolerant quantum error correction provides a strategy to protect information processed by a quantum computer against noise which would otherwise corrupt the data. A fault-tolerant universal quantum computer must implement a universal gate set on the logical level in order to perform arbitrary calculations to in principle unlimited precision. In this paper we characterize the recent demonstration of a fault-tolerant universal gate set in a trapped-ion quantum computer [Postler et al., Nature (London) 605, 7911 (2022)] and identify aspects to improve the design of experimental setups to reach an advantage of logical over physical qubit operation. We show that various criteria to assess the break-even point for fault-tolerant quantum operations are within reach for the ion trap quantum computing architecture under consideration. Furthermore, we analyze the influence of crosstalk in entangling gates for logical state preparation circuits. These circuits can be designed to respect fault tolerance for specific microscopic noise models. We find that an experimentally informed depolarizing noise model captures the essential noise dynamics of the fault-tolerant experiment that we consider, and crosstalk is negligible in the currently accessible regime of physical error rates. For deterministic Pauli state preparation, we provide a fault-tolerant unitary logical qubit initialization circuit, which can be realized without in-sequence measurement and feed-forward of classical information. Additionally, we show that nondeterministic state preparation schemes, i.e., repeat until success, for logical Pauli and magic states perform with higher logical fidelity over their deterministic counterparts for the current and anticipated future regime of physical error rates. Our results offer guidance on improvements of physical qubit operations and validate the experimentally informed noise model as a tool to predict logical failure rates in quantum computing architectures based on trapped ions.

Non-Gaussian quantum state generation by multi-photon subtraction at the telecommunication wavelength.

In the field of continuous-variable quantum information processing, non-Gaussian states with negative values of the Wigner function are crucial for the development of a fault-tolerant universal quantum computer. While several non-Gaussian states have been generated experimentally, none have been created using ultrashort optical wave packets, which are necessary for high-speed quantum computation, in the telecommunication wavelength band where mature optical communication technology is available. In this paper, we present the generation of non-Gaussian states on wave packets with a short 8-ps duration in the 1545.32 nm telecommunication wavelength band using photon subtraction up to three photons. We used a low-loss, quasi-single spatial mode waveguide optical parametric amplifier, a superconducting transition edge sensor, and a phase-locked pulsed homodyne measurement system to observe negative values of the Wigner function without loss correction up to three-photon subtraction. These results can be extended to the generation of more complicated non-Gaussian states and are a key technology in the pursuit of high-speed optical quantum computation.

Noise tailoring for robust amplitude estimation

A universal fault-tolerant quantum computer holds the promise to speed up computational problems that are otherwise intractable on classical computers; however, for the next decade or so, our access is restricted to noisy intermediate-scale quantum (NISQ) computers and, perhaps, early fault tolerant (EFT) quantum computers. This motivates the development of many near-term quantum algorithms including robust amplitude estimation (RAE), which is a quantum-enhanced algorithm for estimating expectation values. One obstacle to using RAE has been a paucity of ways of getting realistic error models incorporated into this algorithm. So far the impact of device noise on RAE is incorporated into one of its subroutines as an exponential decay model, which is unrealistic for NISQ devices and, maybe, for EFT devices; this hinders the performance of RAE. Rather than trying to explicitly model realistic noise effects, which may be infeasible, we circumvent this obstacle by tailoring device noise using randomized compiling to generate an effective noise model, whose impact on RAE closely resembles that of the exponential decay model. Using noisy simulations, we show that our noise-tailored RAE algorithm is able to regain improvements in both bias and precision that are expected for RAE. Additionally, on IBM’s quantum computer ibmq_belem our algorithm demonstrates advantage over the standard estimation technique in reducing bias. Thus, our work extends the feasibility of RAE on NISQ computers, consequently bringing us one step closer towards achieving quantum advantage using these devices.

Low-overhead code concatenation approaches for universal quantum computation

Fault-tolerant quantum circuit design can be done by using a set of transversal gates. However, as there is no quantum error correction code with a universal set of transversal gates, several approaches have been proposed which, in combination of transversal gates, make universal fault-tolerant quantum computation possible. Magic state distillation, gauge fixing, code switching, code concatenation and pieceable fault tolerance are well-known examples of such approaches. However, the overhead of these approaches is one of the main bottlenecks for large-scale quantum computation. In this paper, two approaches for universal fault-tolerant quantum computation, mainly based on code concatenation, are proposed. The first approach combines the code concatenation approach with code switching, pieceable fault tolerance or magic state distillation and the second approach extends the nonuniformity of the concatenated codes by allowing to apply CNOT gates between different codes. The proposed approaches outperform the code concatenation approach in terms of both number of qubits and code distance and have also significantly less resource overhead than magic state distillation. This is achieved at the cost of reducing the effective distance of the concatenated code for implementing non-transversal gates.

Detecting quantum phase transition via magic resource in the XY spin model

Quantum phase transition in the $XY$ spin model with three-spin interaction is investigated using magic resource (non-stabilizerness), which is crucial in universal fault-tolerant quantum computation. The magic quantifier we employ here is defined straightforwardly via characteristic functions of quantum states, which are well defined for all dimensional quantum systems (in sharp contrast to those defined by discrete Wigner functions) and can be easily calculated. We show that the magic quantifier of both the reduced single-site spins and two-site spins of the system ground state increase to their maximum around the critical points for quantum phase transition. This indicates that the magic resource can be used to detect the critical phenomena in the $XY$ spin model and reveals a connection between quantum phase transition in many-body systems and quantum resource in stabilizer quantum computation.

Optimal quantum reservoir computing for the noisy intermediate-scale quantum era.

Universal fault-tolerant quantum computers require millions of qubits with low error rates. Since this technology is years ahead, noisy intermediate-scale quantum (NISQ) computation is receiving tremendous interest. In this setup, quantum reservoir computing is a relevant machine learning algorithm. Its simplicity of training and implementation allows to perform challenging computations on today's available machines. In this Letter, we provide a criterion to select optimal quantum reservoirs, requiring few and simple gates. Our findings demonstrate that they render better results than other commonly used models with significantly less gates and also provide insight on the theoretical gap between quantum reservoir computing and the theory of quantum states' complexity.

Experimental Implementation of Quantum Algorithm for Association Rules Mining

Recently, a quantum algorithm for a fundamentally important task in data mining, association rules mining (ARM), called qARM for short, has been proposed. Notably, qARM achieves significant speedup over its classical counterpart for implementing the main task of ARM, i.e., finding frequent itemsets from a transaction database. In this paper, we experimentally implement qARM on both real quantum computers and a quantum computing simulator via the IBM quantum computing platform. In the first place, we design quantum circuits of qARM for a 2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> 2 transaction database (i.e., a transaction database involving two transactions and two items), and run it on four real five-qubit IBM quantum computers as well as on the simulator. For a larger 4 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> 4 transaction database which would lead to circuits with more qubits and a higher depth than the currently accessible IBM real quantum devices can handle, we also construct the quantum circuits of qARM and execute them on “aer_simulator” alone. Both experimental results show that all the frequent itemsets from the two transaction databases are successfully derived as desired, demonstrating the correctness and feasibility of qARM. Our work may serve as a benchmarking, and provide prototypes for implementing qARM for larger transaction databases on both noisy intermediate-scale quantum devices and universal fault-tolerant quantum computers.

Achieving Fault Tolerance on Capped Color Codes with Few Ancillas

Attaining fault tolerance while maintaining low overhead is one of the main challenges in a practical implementation of quantum circuits. One major technique that can overcome this problem is the flag technique, in which high-weight errors arising from a few faults can be detected by a few ancillas and distinguished using subsequent syndrome measurements. The technique can be further improved using the fact that for some families of codes, errors of any weight are logically equivalent if they have the same syndrome and weight parity, as previously shown in [Phys. Rev. A 104, 042410 (2021)]. In this work, we develop a notion of distinguishable fault set which captures both concepts of flags and weight parities, and extend the use of weight parities in error correction from [Phys. Rev. A 104, 042410 (2021)] to families of capped and recursive capped color codes. We also develop fault-tolerant protocols for error correction, measurement, state preparation, and logical T gate implementation via code switching, which are sufficient for performing fault-tolerant Clifford computation on a capped color code, and performing fault-tolerant universal quantum computation on a recursive capped color code. Our protocols for a capped or a recursive capped color code of any distance require only 2 ancillas, assuming that the ancillas can be reused. The concept of distinguishable fault set also leads to a generalization of the definitions of fault-tolerant gadgets proposed by Aliferis, Gottesman, and Preskill.

Blueprint for a High-Performance Fluxonium Quantum Processor

The technological development of hardware heading toward universal fault-tolerant quantum computation requires a large-scale processing unit with high performance. While fluxonium qubits are promising with high coherence and large anharmonicity, their scalability has not been systematically explored. In this work, we propose a superconducting quantum information processor based on compact high-coherence fluxoniums with suppressed crosstalk, reduced design complexity, improved operational efficiency, high-fidelity gates, and resistance to parameter fluctuations. In this architecture, the qubits are readout dispersively using individual resonators connected to a common bus and manipulated via combined on-chip RF and DC control lines, both of which can be designed to have low crosstalk. A multi-path coupling approach enables exchange interactions between the high-coherence computational states and at the same time suppresses the spurious static ZZ rate, leading to fast and high-fidelity entangling gates. We numerically investigate the cross resonance controlled-NOT and the differential AC-Stark controlled-Z operations, revealing low gate error for qubit-qubit detuning bandwidth of up to 1 GHz. Our study on frequency crowding indicates high fabrication yield for quantum processors consisting of over thousands of qubits. In addition, we estimate low resource overhead to suppress logical error rate using the XZZX surface code. These results promise a scalable quantum architecture with high performance for the pursuit of universal quantum computation.

Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories

Universal fault-tolerant quantum computers will require the use of efficient protocols to implement encoded operations necessary in the execution of algorithms. In this work, we show how solvers for satisfiability modulo theories (SMT solvers) can be used to automate the construction of Clifford circuits with certain fault-tolerance properties and we apply our techniques to a fault-tolerant magic-state-preparation protocol. Part of the protocol requires converting magic states encoded in the color code to magic states encoded in the surface code. Since the teleportation step involves decoding a color code merged with a surface code, we develop a decoding algorithm that is applicable to such codes.

Demonstration of fault-tolerant universal quantum gate operations.

Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error-correcting codes1,2. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design3-5, which, in general, increases the complexity of the implementation. Here we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In particular, we make use of the recently introduced paradigm of flag fault tolerance, where the absence or presence of dangerous errors is heralded by the use of auxiliary flag qubits6-10. We perform a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code11,12, and fault-tolerantly prepare a logical magic state8,13. We then realize a fault-tolerant logical T gate by injecting the magic state by teleportation from one logical qubit onto the other14. We observe the hallmark feature of fault tolerance-a superior performance compared with a non-fault-tolerant implementation. In combination with recently demonstrated repeated quantum error-correction cycles15,16, these results provide a route towards error-corrected universal quantum computation.

Many-Body Quantum Magic

Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation. To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic. Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions). Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangled, with focus on the important case of many qubits, at a quantitative level. We first address the basic question of how "magical" a many-body state can be, and show that the maximum magic of an $n$-qubit state is essentially $n$, simultaneously for a range of "good" magic measures. We then show that, in fact, almost all $n$-qubit pure states have magic of nearly $n$. In the quest for explicit, scalable cases of highly entangled states whose magic can be understood, we connect the magic of hypergraph states with the second-order nonlinearity of their underlying Boolean functions. Next, we go on and investigate many-body magic in practical and physical contexts. We first consider a variant of MBQC where the client is restricted to Pauli measurements, in which magic is a necessary feature of the initial "resource" state. We show that $n$-qubit states with nearly $n$ magic, or indeed almost all states, cannot supply nontrivial speedups over classical computers. We then present an example of analyzing the magic of "natural" condensed matter systems of physical interest. We apply the Boolean function techniques to derive explicit bounds on the magic of certain representative 2D SPT states, and comment on possible further connections between magic and the quantum complexity of phases of matter.

Constraints on magic state protocols from the statistical mechanics of Wigner negativity

Magic states are key ingredients in schemes to realize universal fault-tolerant quantum computation. Theories of magic states attempt to quantify this computational element via monotones and determine how these states may be efficiently transformed into useful forms. Here, we develop a statistical mechanical framework based on majorization to describe Wigner negative magic states for qudits of odd prime dimension processed under Clifford circuits. We show that majorization allows us to both quantify disorder in the Wigner representation and derive upper bounds for magic distillation. These bounds are shown to be tighter than other bounds, such as from mana and thauma, and can be used to incorporate hardware physics, such as temperature dependence and system Hamiltonians. We also show that a subset of single-shot Rényi entropies remain well-defined on quasi-distributions, are fully meaningful in terms of data processing and can acquire negative values that signal magic. We find that the mana of a magic state is the measure of divergence of these Rényi entropies as one approaches the Shannon entropy for Wigner distributions, and discuss how distillation lower bounds could be obtained in this setting. This use of majorization for quasi-distributions could find application in other studies of non-classicality, and raises nontrivial questions in the context of classical statistical mechanics.

Generation of Schrödinger cat states with Wigner negativity using a continuous-wave low-loss waveguide optical parametric amplifier.

Continuous-wave (CW) squeezed light is used in the generation of various optical quantum states, and thus is a fundamental resource of fault-tolerant universal quantum computation using optical continuous variables. To realize a practical quantum computer, a waveguide optical parametric amplifier (OPA) is an attractive CW squeezed light source in terms of its THz-order bandwidth and suitability for modularization. The usages of a waveguide OPA in quantum applications thus far, however, are limited due to the difficulty of the generation of the squeezed light with a high purity. In this paper, we report the first observation of Wigner negativity of the states generated by a heralding method using a waveguide OPA. We generate Schrödinger cat states at the wavelength of 1545 nm with Wigner negativity using a quasi-single-mode ZnO-doped periodically poled LiNbO3 waveguide module we developed. Wigner negativity is regarded as an important indicator of the usefulness of the quantum states as it is essential in the fault-tolerant universal quantum computation. Our result shows that our waveguide OPA can be used in wide range of quantum applications leading to a THz-clock optical quantum computer.