Quantum annealers require accurate control and optimized operation schemes to reduce noise levels, in order to eventually demonstrate a computational advantage over classical algorithms. We study a high coherence four-junction capacitively shunted flux qubit (CSFQ), using dispersive measurements to extract system parameters and model the device. Josephson junction asymmetry inherent to the device causes a deleterious nonlinear cross-talk when annealing the qubit. We implement a nonlinear annealing path to correct the asymmetry in-situ, resulting in a substantial increase in the probability of the qubit being in the correct state given an applied flux bias. We also confirm the multi-level structure of our CSFQ circuit model by annealing it through small spectral gaps and observing quantum signatures of energy level crossings. Our results demonstrate an anneal-path correction scheme designed and implemented to improve control accuracy for high-coherence and high-control quantum annealers, which leads to an enhancement of success probability in annealing protocols.
We propose a novel architecture for superconducting circuits to improve the efficiency of a quantum annealing system. To increase the capability of a circuit, it is desirable for a qubit to be coupled not only with adjacent qubits but also with other qubits located far away. We introduce a circuit that uses a lumped element resonator coupled each with one qubit. The resonator-qubit pairs are coupled by rf-superconducting quantum interference device (SQUID) based couplers. These pairs make a large quantum system for quantum annealer. This system could prepare the problem Hamiltonian and tune th…
Scaling the number of qubits while maintaining high-fidelity quantum gates remains a key challenge for quantum computing. Presently, superconducting quantum processors with >50-qubits are actively available. For such systems, fixed-frequency transmons are attractive due to their long coherence and noise immunity. However, scaling fixed-frequency architectures proves challenging due to precise relative frequency requirements. Here we employ laser annealing to selectively tune transmon qubits into desired frequency patterns. Statistics over hundreds of annealed qubits demonstrate an empirical tu…
We propose a quantum annealing protocol that more effectively probes the dynamics of a single qubit on D-Wave's quantum annealing hardware. This protocol uses D-Wave's h-gain schedule functionality, which allows the rapid quenching of the longitudinal magnetic field at arbitrary points during the anneal. This features enables us to distinguish between open and closed system dynamics as well as the presence or absence of longitudinal magnetic field noise. We show that both thermal and magnetic field fluctuations are key sources of noise that need to be included in an open quantum system model t…
The rotational and fine structure of open-shell molecules in a $Σ$ electronic state gives rise to crossings between Zeeman states of different parity. These crossings become avoided in the presence of an electric field. We propose an algorithm that encodes Ising models into qubits defined by pairs of $^2Σ$ molecules sharing an excitation near these avoided crossings. This can be used to realize a transverse field Ising model tunable by an external electric or magnetic field, suitable for quantum annealing applications. We perform dynamical calculations for several examples with one- and two-di…
Preparing ground states of Hamiltonians is important in the condensed matter physics and the quantum chemistry. The interaction Hamiltonians typically contain not only diagonal but also off-diagonal elements. Although quantum annealing provides a way to prepare a ground state of a Hamiltonian, we can only use the Hamiltonian with Ising interaction by using currently available commercial quantum annealing devices. In this work, we propose a quantum annealing for the XXZ model, which contains both Ising interaction and energy-exchange interaction, by using inductively coupled superconducting flu…
Quantum technology is maturing to the point where quantum devices, such as quantum communication systems, quantum random number generators and quantum simulators, may be built with capabilities exceeding classical computers. A quantum annealer, in particular, solves hard optimisation problems by evolving a known initial configuration at non-zero temperature towards the ground state of a Hamiltonian encoding a given problem. Here, we present results from experiments on a 108 qubit D-Wave One device based on superconducting flux qubits. The strong correlations between the device and a simulated …
An important challenge in superconducting quantum computing is the need to physically couple many devices using quasi-two-dimensional fabrication processes. Recent advances in the design and fabrication of quantum annealing processors have enabled an increase in pairwise connectivity among thousands of qubits. One benefit of this is the ability to minor-embed optimization problems using fewer physical qubits for each logical spin. Here we demonstrate the benefit of this progress in the problem of minimizing the energy of three-dimensional spin glasses. Comparing the previous generation D-Wave …
We propose a two qubit experiment for validating tunable antiferromagnetic $XX$ interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be non-stoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic $XX$ interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state.…
Nested quantum annealing correction (NQAC) is an error correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. The encoding replaces each logical qubit by a complete graph of degree $C$. The nesting level $C$ represents the distance of the error-correcting code and controls the amount of protection against thermal and control errors. Theoretical mean-field analyses and empirical data obtained with a D-Wave Two quantum annealer (supporting up to $512$ qubits) showed that NQAC has the potential to achieve a scal…
Quantum annealing may provide advantages over simulated annealing on solving some problems such as Kth order binary optimization problem. No feasible architecture exists to implement the high-order optimization problem (K > 2) on current quantum annealing hardware. We propose a two-dimensional quantum annealing architecture to solve the 4th order binary optimization problem by encoding four-qubit interactions within the coupled local fields acting on a set of physical qubits. All possible four-body coupling terms for an N-qubit system can be implemented through this architecture and are readil…
The real-time flux dynamics of up to three superconducting quantum interference devices (SQUIDs) are studied by numerically solving the time-dependent Schrödinger equation. The numerical results are used to scrutinize the mapping of the flux degrees of freedom onto two-level systems (the qubits) as well as the performance of the intermediate SQUID as a tunable coupling element. It is shown that the qubit representation yields a good description of the flux dynamics during quantum annealing and the presence of the tunable coupling element does not have negative effects on the overall performanc…
Quantum information processing offers dramatic speedups, yet is famously susceptible to decoherence, the process whereby quantum superpositions decay into mutually exclusive classical alternatives, thus robbing quantum computers of their power. This has made the development of quantum error correction an essential and inescapable aspect of both theoretical and experimental quantum computing. So far little is known about protection against decoherence in the context of quantum annealing, a computational paradigm which aims to exploit ground state quantum dynamics to solve optimization problems …
Experimental Measurement and Theoretical Analysis of Energy Relaxation Rates of an Interacting Two-Qubit System on a D-Wave Quantum Annealer
Tokyo University of Science（东京科学大学）；RIKEN（理化学研究所）；Department of Electrical, Electronic, and Communication Engineering, Faculty of Science and Engineering, Chuo University（中央大学理工学部电气电子通信工程系）
The development of quantum annealing machines (QAMs) based on superconducting qubits has progressed greatly in recent years and these machines are now widely used in both academia and commerce. On the other hand, QAMs based on semiconductor nanostructures such as quantum dots (QDs) appear to be still at the initial elementary research stage because of difficulty in controlling the interaction between qubits. In this paper, we review a QAM based on a semiconductor nanostructures such as floating gates (FGs) or QDs from the viewpoint of the integration of qubits. We theoretically propose the use…
Algorithms for searching and sorting data sets on quantum annealing systems are presented. Search algorithms for unordered data sets are developed. A sorting algorithm for data sets is provided, with a consideration of sort stability. Scalability of the algorithms, considering both the number of qubits required and the qubit connectivity, is characterized as a function of problem size.
A significant challenge in quantum annealing is to map a real-world problem onto a hardware graph of limited connectivity. If the maximum degree of the problem graph exceeds the maximum degree of the hardware graph, one employs minor embedding in which each logical qubit is mapped to a tree of physical qubits. Pairwise interactions between physical qubits in the tree are set to be ferromagnetic with some coupling strength $F<0$. Here we address the question of what value $F$ should take in order to maximise the probability that the annealer finds the correct ground-state of an Ising problem. T…
We present a general error-correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. Given any Ising model optimization problem, the encoding replaces each logical qubit by a complete graph of degree $C$, representing the distance of the error-correcting code. A subsequent minor-embedding step then implements the encoding on the underlying hardware graph of the quantum annealer. We demonstrate experimentally that the performance of a D-Wave Two quantum annealing device improves as $C$ grows. We show that the per…
Quantum annealing is an optimization technique which potentially leverages quantum tunneling to enhance computational performance. Existing quantum annealers use superconducting flux qubits with short coherence times, limited primarily by the use of large persistent currents $I_\mathrm{p}$. Here, we examine an alternative approach, using qubits with smaller $I_\mathrm{p}$ and longer coherence times. We demonstrate tunable coupling, a basic building block for quantum annealing, between two flux qubits with small ($\sim 50~\mathrm{nA}$) persistent currents. Furthermore, we characterize qubit coh…
For quantum annealing, as opposed to circuit based quantum computing, the solution to a computational problem is encoded in the ground state of a quantum system. Therefore its susceptibility to environmental effects is a different but not less important open question essential to a scalable implementation. In this work we use renormalization group techniques to study the effect of Ohmic baths in a regime typical for practical adiabatic quantum computation. We show qualitative change to the effective Hamiltonian as well as the reduced qubit density matrix encoding entanglement between system an…
We propose a realizable circuit QED architecture for engineering states of a superconducting resonator off-resonantly coupled to an ancillary superconducting qubit. The qubit-resonator dispersive interaction together with a microwave drive applied to the qubit gives rise to a Kerr resonator with two-photon driving that enables us to efficiently engineer the quantum state of the resonator such as generation of the Schrodinger cat states for resonator-based universal quantum computation. Moreover, the presented architecture is easily scalable for solving optimization problem mapped into the Isin…
Decoherence of a tunable capacitively shunted flux qubit
Department of Physics and Astronomy, University of Waterloo（物理与天文学系，滑铁卢大学）；Department of Electrical and Computer Engineering, University of Waterloo（电气与计算机工程系，滑铁卢大学）；Department of Physics and Astronomy, Dartmouth College（物理与天文学系，达特茅斯学院）；Research Laboratory of Electronics, Massachusetts Institute of Technology（电子学研究实验室，麻省理工学院）
We study a bifurcation mechanism of quantum annealing. Using spins with quantum number S=1, we construct a simple model to make a bifurcation. The qutrit can be composed by nesting two qubits. We numerically solve the Schroedinger equation to confirm that the bifurcation-based quantum annealing (BQA) works well and the ground state can be found efficiently. The result is compared with that by the standard quantum annealing (QA) using qubits. We find that the performance of the BQA is comparable to the standard QA, or gives better results in some cases.
Quantum simulation has emerged as a valuable arena for demonstrating and understanding the capabilities of near-term quantum computers. Quantum annealing has been used successfully in simulating a range of open quantum systems, both at equilibrium and out of equilibrium. However, in all previous experiments, annealing has been too slow to simulate a closed quantum system coherently, due to the onset of thermal effects from the environment. Here we demonstrate coherent evolution through a quantum phase transition in the paradigmatic setting of the 1D transverse-field Ising chain, using up to 20…
As a wide variety of quantum computing platforms become available, methods for assessing and comparing the performance of these devices are of increasing interest and importance. Inspired by the success of single-qubit error rate computations for tracking the progress of gate-based quantum computers, this work proposes a Quantum Annealing Single-qubit Assessment (QASA) protocol for quantifying the performance of individual qubits in quantum annealing computers. The proposed protocol scales to large quantum annealers with thousands of qubits and provides unique insights into the distribution of…
Temporal information processing on a 4,500-qubit quantum annealer
Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) UIB-CSIC（跨学科物理与复杂系统研究所（IFISC） UIB-CSIC）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；USRA Research Institute for Advanced Computer Science (RIACS)（美国大学空间研究协会先进计算机科学研究院（RIACS））；NEST, Scuola Normale Superiore（NEST，比萨高等师范学院）；D-Wave Quantum（D-Wave量子）；New Mexico Consortium（新墨西哥联盟）
Various sources of noise limit the performance of quantum computers by altering qubit states in an uncontrolled manner throughout computations and reducing their coherence time. In quantum annealers, this noise introduces additional fluctuations to the parameters defining the original problem Hamiltonian, such that they find the ground states of problems perturbed from those originally programmed. Here we describe a method to benchmark the amount of noise affecting the programmed Hamiltonian of a quantum annealer. We show that a sequence of degenerate runs with the coefficients of the programm…
Quantum computing is no longer a nascent field. Programmable quantum annealing devices with more that 1000 qubits are commercially available. How does one know that a putative quantum annealing device is indeed quantum? How should one go about benchmarking its performance and compare it to classical algorithms? How can its performance be improved by error correction? In this contribution to the focus collection on "What would you do with 1000 qubits?", we review the work we and others have done in this area, since the first D-Wave quantum annealer with 108 qubits was made available to us. Some…
We show a significant reduction of errors for an architecture of quantum annealers (QA) where bosonic modes mediate the interaction between qubits. These systems have a large redundancy in the subspace of solutions, supported by arbitrarily large bosonic occupations. We explain how this redundancy leads to a mitigation of errors when the bosonic modes operate in the ultrastrong coupling regime. Numerical simulations also predict a large increase of qubit coherence for a specific annealing problem with mediated interactions. We provide evidences that noise reduction occurs in more general types…
(Withdrawn) This work describes an efficient user-side method of calibrating and correcting quantum annealing computers. For quantum annealing computers based on the Ising model, the method measures the residual bias of the h and J coefficients. Once measured, these biases can then be nulled in subsequent runs for any problem of interest. This method also returns a temperature for each qubit based on the measured versus the expected qubit distributions computed from a Boltzmann distribution model.
Quantum annealing has recently been used to determine the Ramsey numbers R(m,2) for 3 < m < 9 and R(3,3) [Bian et al. (2013) PRL 111, 130505]. This was greatly celebrated as the largest experimental implementation of an adiabatic evolution algorithm to that date. However, in that computation, more than 66% of the qubits used were auxiliary qubits, so the sizes of the Ramsey number Hamiltonians used were tremendously smaller than the full 128-qubit capacity of the device used. The reason these auxiliary qubits were needed was because the best quantum annealing devices at the time (and still now…
Solving optimization problems on quantum annealers usually requires each variable of the problem to be represented by a connected set of qubits called a logical qubit or a chain. Chain weights, in the form of ferromagnetic coupling between the chain qubits, are applied so that the physical qubits in a chain favor taking the same value in low energy samples. Assigning a good chain-strength value is crucial for the ability of quantum annealing to solve hard problems, but there are no general methods for computing such a value and, even if an optimal value is found, it may still not be suitable b…
We present \emph{multi-qubit correction} (MQC) as a novel postprocessing method for quantum annealers that views the evolution in an open-system as a Gibbs sampler and reduces a set of excited states to a new synthetic state with lower energy value. After sampling from the ground state of a given (Ising) Hamiltonian, MQC compares pairs of excited states to recognize virtual tunnels--i.e., a group of qubits that changing their states simultaneously can result in a new state with lower energy value--and successively converges to the ground state. Experimental results using D-Wave 2000Q quantum a…
Solving Differential Equations Using Continuous-Variable Quantum Annealing
Chuo University（中央大学）；The University of Tokyo（东京大学）；Ochanomizu University（御茶水女子大学）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
In this letter we present an efficient gap-independent cooling scheme for a quantum annealer that benefits from finite temperatures. We choose a system based on superconducting flux qubits as a prominent example of current quantum annealing platforms. We propose coupling the qubit system transversely to a coplanar waveguide to counter noise and heating that arise from always-present longitudinal thermal noise. We provide a schematic circuit layout for the system and show how, for feasible coupling strengths, we achieve global performance enhancements. Specifically, we achieve cooling improveme…
As superconducting quantum processors increase in complexity, techniques to overcome constraints on frequency crowding are needed. The recently developed method of laser-annealing provides an effective post-fabrication method to adjust the frequency of superconducting qubits. Here, we present an automated laser-annealing apparatus based on conventional microscopy components and demonstrate preservation of highly coherent transmons. In one case, we observe a two-fold increase in coherence after laser-annealing and perform noise spectroscopy on this qubit to investigate the change in defect feat…
Quantum annealing allows for quantum fluctuations to be used used to assist in finding the solution to some of the worlds most challenging computational problems. Recently, this field has attracted much interest because of the construction of large-scale flux-qubit based quantum annealing devices. There has been recent work on [Chancellor NJP 19(2):023024, 2017] how the control protocols of these devices can be modified so that individual annealer calls on real devices can take initial conditions. Development is being undertaken to implement such protocols in the quantum annealing devices desi…
To gain better insight into the complexity theory of quantum annealing, we propose and solve a class of spin systems which contain bottlenecks of the kind expected to dominate the runtime of quantum annealing as it tries to solve difficult optimization problems. We uncover a noise amplification effect at these bottlenecks, whereby tunneling rates caused by flux-qubit noise scale in proportion to the number of qubits $N$ in the limit that $N\to \infty$. By solving the incoherent annealing dynamics exactly, we find a wide range of regimes where the probability that a quantum annealer remains in …
We benchmark the quantum processing units of the largest quantum annealers to date, the 5000+ qubit quantum annealer Advantage and its 2000+ qubit predecessor D-Wave 2000Q, using tail assignment and exact cover problems from aircraft scheduling scenarios. The benchmark set contains small, intermediate, and large problems with both sparsely connected and almost fully connected instances. We find that Advantage outperforms D-Wave 2000Q for almost all problems, with a notable increase in success rate and problem size. In particular, Advantage is also able to solve the largest problems with 120 lo…
Quantum annealers of D-Wave Systems, Inc., offer an efficient way to compute high quality solutions of NP-hard problems. This is done by mapping a problem onto the physical qubits of the quantum chip, from which a solution is obtained after quantum annealing. However, since the connectivity of the physical qubits on the chip is limited, a minor embedding of the problem structure onto the chip is required. In this process, and especially for smaller problems, many qubits will stay unused. We propose a novel method, called parallel quantum annealing, to make better use of available qubits, where…
We argue that a complete description of quantum annealing (QA) implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquastic terms in the effective quantum Ising Hamiltonians that are typically used to describe QA with flux-qubits. We explicitly demonstrate the effect of these geometric interactions when QA is performed with a system of one and two coupled flux-qubits. The realization of non-stoquastic …
Diabatic quantum annealing (DQA) is an alternative algorithm to adiabatic quantum annealing (AQA) that can be used to circumvent the exponential slowdown caused by small minima in the annealing energy spectrum. We present the locally suppressed transverse-field (LSTF) protocol, a heuristic method for making stoquastic optimization problems compatible with DQA. We show that, provided an optimization problem intrinsically has magnetic frustration due to inhomogeneous local fields, a target qubit in the problem can always be manipulated to create a double minimum in the energy gap between the gro…
Optimization of Connection Patterns between Mobile Phones and Base Stations using Quantum Annealing
Research Institute of Advanced Technology, SoftBank Corp.（软银公司先进技术研究院）；Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究院）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Sigma-I Co., Ltd.（Sigma-I有限公司）
We demonstrate, for the first time, that a quantum flux parametron (QFP) is capable of acting as both isolator and amplifier in the readout circuit of a capacitively shunted flux qubit (CSFQ). By treating the QFP like a tunable coupler and biasing it such that the coupling is off, we show that $T_1$ of the CSFQ is not impacted by Purcell loss from its low-Q readout resonator ($Q_e = 760$) despite being detuned by only $40$ MHz. When annealed, the QFP amplifies the qubit's persistent current signal such that it generates a flux qubit-state-dependent frequency shift of $85$ MHz in the readout re…
Quantum annealing (QA) provides us with a way to solve combinatorial optimization problems. In the previous demonstration of the QA, a superconducting flux qubit (FQ) was used. However, the flux qubits in these demonstrations have a short coherence time such as tens of nano seconds. For the purpose to utilize quantum properties, it is necessary to use another qubit with a better coherence time. Here, we propose to use a capacitive-shunted flux qubit (CSFQ) for the implementation of the QA. The CSFQ has a few order of magnitude better coherence time than the FQ used in the QA. We theoretically …
Advancements in computing based on qubit networks, and in particular the flux-qubit processor architecture developed by D-Wave System's Inc., have enabled the physical simulation of quantum-dot cellular automata (QCA) networks beyond the limit of classical methods. However, the embedding of QCA networks onto the available processor architecture is a key challenge in preparing such simulations. In this work, two approaches to embedding QCA circuits are characterized: a dense placement algorithm that uses a routing method based on negotiated congestion; and a heuristic method implemented in D-Wa…
Perturbative anticrossings have long been identified as a potential computational bottleneck for quantum annealing. This bottleneck can appear, for example, when a uniform transverse driver Hamiltonian is applied to each qubit. Previous theoretical research sought to alleviate such anticrossings by adjusting the transverse driver Hamiltonians on individual qubits according to a perturbative approximation. Here we apply this principle to a physical implementation of quantum annealing in a D-Wave 2000Q system. We use samples from the quantum annealing hardware and per-qubit anneal offsets to pro…
