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Transitioning from quantum computation on physical qubits to quantum computation on encoded, logical qubits can improve the error rate of operations, and will be essential for realizing valuable quantum computational advantages. Using a neutral atom quantum processor with 256 qubits, each an individual Ytterbium atom, we demonstrate the entanglement of 24 logical qubits using the distance-two [[4,2,2]] code, simultaneously detecting errors and correcting for lost qubits. We also implement the Bernstein-Vazirani algorithm with up to 28 logical qubits encoded in the [[4,1,2]] code, showing better-than-physical error rates. We demonstrate fault-tolerant quantum computation in our approach, guided by the proposal of Gottesman (2016), by performing repeated loss correction for both structured and random circuits encoded in the [[4,2,2]] code. Finally, since distance-two codes can correct qubit loss, but not other errors, we show repeated loss and error correction using the distance-three [[9,1,3]] Bacon-Shor code. These results begin to clear a path for achieving scientific quantum advantage with a programmable neutral atom quantum processor., Comment: 17 pages, 16 figures

In this work, we introduce a new iterative quantum algorithm, called Iterative Symphonic Tunneling for Satisfiability problems (IST-SAT), which solves quantum spin glass optimization problems using high-frequency oscillating transverse fields. IST-SAT operates as a sequence of iterations, in which bitstrings returned from one iteration are used to set spin-dependent phases in oscillating transverse fields in the next iteration. Over several iterations, the novel mechanism of the algorithm steers the system toward the problem ground state. We benchmark IST-SAT on sets of hard MAX-3-XORSAT problem instances with exact state vector simulation, and report polynomial speedups over trotterized adiabatic quantum computation (TAQC) and the best known semi-greedy classical algorithm. When IST-SAT is seeded with a sufficiently good initial approximation, the algorithm converges to exact solution(s) in a polynomial number of iterations. Our numerical results identify a critial Hamming radius(CHR), or quality of initial approximation, where the time-to-solution crosses from exponential to polynomial scaling in problem size. By combining IST-SAT with future classical or quantum approximation algorithms, larger gains may be achieved. The mechanism we present in this work thus presents a new path toward achieving quantum advantage in optimization.

In this work we study the properties of dissipatively stabilized steady states of noisy quantum algorithms, exploring the extent to which they can be well approximated as thermal distributions, and proposing methods to extract the effective temperature T. We study an algorithm called the Relaxational Quantum Eigensolver (RQE), which is one of a family of algorithms that attempt to find ground states and balance error in noisy quantum devices. In RQE, we weakly couple a second register of auxiliary "shadow" qubits to the primary system in Trotterized evolution, thus engineering an approximate zero-temperature bath by periodically resetting the auxiliary qubits during the algorithm's runtime. Balancing the infinite temperature bath of random gate error, RQE returns states with an average energy equal to a constant fraction of the ground state. We probe the steady states of this algorithm for a range of base error rates, using several methods for estimating both T and deviations from thermal behavior. In particular, we both confirm that the steady states of these systems are often well-approximated by thermal distributions, and show that the same resources used for cooling can be adopted for thermometry, yielding a fairly reliable measure of the temperature. These methods could be readily implemented in near-term quantum hardware, and for stabilizing and probing Hamiltonians where simulating approximate thermal states is hard for classical computers., Comment: 14 pages, 7 Figures

Reservoir engineering is a powerful technique to autonomously stabilize a quantum state. Traditional schemes involving multi-body states typically function for discrete entangled states. In this work, we enhance the stabilization capability to a continuous manifold of states with programmable stabilized state selection using multiple continuous tuning parameters. We experimentally achieve $84.6\%$ and $82.5\%$ stabilization fidelity for the odd and even-parity Bell states as two special points in the manifold. We also perform fast dissipative switching between these opposite parity states within $1.8\mu s$ and $0.9\mu s$ by sequentially applying different stabilization drives. Our result is a precursor for new reservoir engineering-based error correction schemes.

A canonical feature of the constraint satisfaction problems in NP is approximation hardness, where in the worst case, finding sufficient-quality approximate solutions is exponentially hard for all known methods. Fundamentally, the lack of any guided local minimum escape method ensures both exact and approximate classical approximation hardness, but the equivalent mechanism(s) for quantum algorithms are poorly understood. For algorithms based on Hamiltonian time evolution, we explore this question through the prototypically hard MAX-3-XORSAT problem class. We conclude that the mechanisms for quantum exact and approximation hardness are fundamentally distinct. We review known results from the literature, and identify mechanisms that make conventional quantum methods (such as Adiabatic Quantum Computing) weak approximation algorithms in the worst case. We construct a family of spectrally filtered quantum algorithms that escape these issues, and develop analytical theories for their performance. We show that, for random hypergraphs in the approximation-hard regime, if we define the energy to be $E = N_{\mathrm{unsat}}-N_{\mathrm{sat}}$, spectrally filtered quantum optimization will return states with $E \leq q_m E_{\mathrm{GS}}$ (where $E_{\rm GS}$ is the ground state energy) in sub-quadratic time, where conservatively, $q_m \simeq 0.59$. This is in contrast to $q_m \to 0$ for the hardest instances with classical searches. We test all of these claims with extensive numerical simulations. We do not claim that this approximation guarantee holds for all possible hypergraphs, though our algorithm's mechanism can likely generalize widely. These results suggest that quantum computers are more powerful for approximate optimization than had been previously assumed., Comment: NOTE: arxiv METADATA abstract abridged compared to the manuscript; 43 pages, 18 figures, 1 table

The exponential suppression of macroscopic quantum tunneling (MQT) in the number of elements to be reconfigured is an essential element of broken symmetry phases. This suppression is also a core bottleneck in quantum algorithms, such as traversing an energy landscape in optimization, and adiabatic state preparation more generally. In this work, we demonstrate exponential acceleration of MQT through Floquet engineering with the application of a uniform, high frequency transverse drive field. Using the ferromagnetic phase of the transverse field Ising model in one and two dimensions as a prototypical example, we identify three phenomenological regimes as a function of drive strength. For weak drives, the system exhibits exponentially decaying tunneling rates but robust magnetic order; in the crossover regime at intermediate drive strength, we find polynomial decay of tunnelling alongside vanishing magnetic order; and at very strong drive strengths both the Rabi frequency and time-averaged magnetic order are approximately constant with increasing system size. We support these claims with extensive full wavefunction and tensor network numerical simulations, and theoretical analysis. An experimental test of these results presents a technologically important and novel scientific question accessible on NISQ-era quantum computers., Comment: 4.5+7 pages, 10 figs, improved presentation throughout

Analog quantum optimization methods, such as quantum annealing, are promising and at least partially noise tolerant ways to solve hard optimization and sampling problems with quantum hardware. However, they have thus far failed to demonstrate broadly applicable quantum speedups, and an important contributing factor to this is slowdowns from embedding, the process of mapping logical variables to long chains of physical qubits, enabling arbitrary connectivity on the short-ranged 2d hardware grid. Beyond the spatial overhead in qubit count, embedding can lead to severe time overhead, arising from processes where individual chains ``freeze" into ferromagnetic states at different times during evolution, and once frozen the tunneling rate of this single logical variable decays exponentially in chain length. We show that this effect can be substantially mitigated by local AC variation of the qubit parameters as in the RFQA protocol (Kapit and Oganesyan, Quant. Sci. Tech. \textbf{6}, 025013 (2021)), through a mechanism we call Symphonic Tunneling. We provide general arguments and substantial numerical evidence to show that AC-driven multi-qubit tunneling is dramatically faster than its DC counterpart, and since ST is not a 1d-specific mechanism, this enhancement should extend to clusters of coupled chains as well. And unlike a uniform transverse field, in higher dimensions this method cannot be efficiently simulated classically. We explore schemes to synchronize the AC tones within chains to further improve performance. Implemented at scale, these methods could significantly improve the prospects for achieving quantum scaling advantages in near-term hardware., Comment: 22 pages, 8 figs

Large-scale quantum computers will inevitably need quantum error correction (QEC) to protect information against decoherence. Given that the overhead of such error correction is often formidable, autonomous quantum error correction (AQEC) proposals offer a promising near-term alternative. AQEC schemes work by transforming error states into excitations that can be efficiently removed through engineered dissipation. The recently proposed AQEC scheme by Li et al., called the Star code, can autonomously correct or suppress all single qubit error channels using two transmons as encoders with a tunable coupler and two lossy resonators as a cooling source. The Star code requires only two-photon interactions and can be realized with linear coupling elements, avoiding experimentally challenging higher-order terms needed in many other AQEC proposals, but needs carefully selected parameters to achieve quadratic improvements in logical states' lifetimes. Here, we theoretically and numerically demonstrate the optimal parameter choices in the Star Code. We further discuss adapting the Star code to other planar superconducting circuits, which offers a scalable alternative to single qubits for incorporation in larger quantum computers or error correction codes.

The analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and is difficult to accomplish in general. Symmetries imply conservation laws, which partition Hilbert space into invariant subspaces of the time-evolution operator, each of which is demarcated according to its conserved quantity. We show that, starting from a chosen basis, any invariant, symmetry-protected subspaces which are diagonal in that basis are discoverable using transitive closure on graphs representing state-to-state transitions under $k$-local unitary operations. Importantly, the discovery of these subspaces relies neither upon the explicit identification of a symmetry operator or its eigenvalues nor upon the construction of matrices of the full Hilbert space dimension. We introduce two classical algorithms, which efficiently compute and elucidate features of these subspaces. The first algorithm explores the entire symmetry-protected subspace of an initial state in time complexity linear to the size of the subspace by closing local basis state-to-basis state transitions. The second algorithm determines, with bounded error, if a given measurement outcome of a dynamically-generated state is within the symmetry-protected subspace of the state in which the dynamical system is initialized. We demonstrate the applicability of these algorithms by performing post-selection on data generated from emulated noisy quantum simulations of three different dynamical systems: the Heisenberg-XXX model and the $T_6$ and $F_4$ quantum cellular automata. Due to their efficient computability and indifference to identifying the underlying symmetry, these algorithms lend themselves to the post-selection of quantum computer data, optimized classical simulation of quantum systems, and the discovery of previously hidden symmetries in quantum mechanical systems., Comment: 23 pages, 7 figures, 4 appendices

Large-scale quantum computers will inevitably need quantum error correction to protect information against decoherence. Traditional error correction typically requires many qubits, along with high-efficiency error syndrome measurement and real-time feedback. Autonomous quantum error correction (AQEC) instead uses steady-state bath engineering to perform the correction in a hardware-efficient manner. We realize an AQEC scheme, implemented with only two transmon qubits in a 2D scalable architecture, that actively corrects single-photon loss and passively suppresses low-frequency dephasing using six microwave drives. Compared to uncorrected encoding, factors of 2.0, 5.1, and 1.4 improvements are experimentally witnessed for the logical zero, one, and superposition states. Our results show the potential of implementing hardware-efficient AQEC to enhance the reliability of a transmon-based quantum information processor.

Adoption of fast, parametric coupling elements has improved the performance of superconducting qubits, enabling recent demonstrations of quantum advantage in randomized sampling problems. The development of low loss, high contrast couplers is critical for scaling up these systems. We present a blueprint for a gate-tunable coupler realized with a two-dimensional electron gas in an InAs/InGaAs heterostructure. Rigorous numerical simulations of the semiconductor and high frequency electromagnetic behavior of the coupler and microwave circuitry yield an on/off ratio of more than one order of magnitude. We give an estimate of the dielectric-limited loss from the inclusion of the coupler in a two qubit system, with coupler coherences ranging from a few to tens of microseconds., Comment: 25 pages, 7 figures

We propose a novel superconducting logical qubit architecture, called the Cold Echo Qubit (CEQ), which is capable of preserving quantum information for much longer timescales than any of its component parts. The CEQ operates fully autonomously, requiring no measurement or feedback, and is compatible with relatively strong interaction elements, allowing for fast, high fidelity logical gates between multiple CEQ's. Its quantum state is protected by a combination of strong interactions and microwave driving, which implements a form of many-body dynamical decoupling to suppress phase noise. Estimates based on careful theoretical analysis and numerical simulations predict improvements in lifetimes and gate fidelities by an order of magnitude or more compared to the current state of the art, assuming no improvements in base coherence. Here, we consider the simplest possible implementation of the CEQ, using a pair of fluxonium qubits shunted through a shared mutual inductance. While not necessarily the best possible implementation, it is the easiest to test experimentally and should display coherence well past breakeven (as compared to the limiting coherence times of its components). A more complex three-node circuit is also presented; it is expected to roughly double the coherence of its two-fluxonium counterpart., Comment: 8 pages including supplemental information, 5 figures

Processing quantum information using quantum three-level systems or qutrits as the fundamental unit is an alternative to contemporary qubit-based architectures with the potential to provide significant computational advantages. We demonstrate a fully programmable two-qutrit quantum processor by utilizing the third energy eigenstates of two transmons. We develop a parametric coupler to achieve excellent connectivity in the nine-dimensional Hilbert space enabling efficient implementations of two-qutrit gates. We characterize our processor by realizing several algorithms like Deutsch-Jozsa, Bernstein-Vazirani, and Grover's search. Our efficient ancilla-free protocols allow us to show that two stages of Grover's amplification can improve the success rates of an unstructured search with quantum advantage. Our results pave the way for building fully programmable ternary quantum processors using transmons as building blocks for a universal quantum computer., Comment: 5 pages, 3 figures in main text, 14 pages in total

Quantum cellular automata (QCA) evolve qubits in a quantum circuit depending only on the states of their neighborhoods and model how rich physical complexity can emerge from a simple set of underlying dynamical rules. For instance, Goldilocks QCA depending on trade-off principles exhibit non-equilibrating coherent dynamics and generate complex mutual information networks, much like the brain. The inability of classical computers to simulate large quantum systems is a hindrance to understanding the physics of quantum cellular automata, but quantum computers offer an ideal simulation platform. Here we demonstrate the first experimental realization of QCA on a digital quantum processor, simulating a one-dimensional Goldilocks rule on chains of up to 23 superconducting qubits. Employing low-overhead calibration and error mitigation techniques, we calculate population dynamics and complex network measures indicating the formation of small-world mutual information networks. Unlike random states, these networks decohere at fixed circuit depth independent of system size; the largest of which corresponds to 1,056 two-qubit gates. Such computations may open the door to the employment of QCA in applications like the simulation of strongly-correlated matter or beyond-classical computational demonstrations.

We introduce a simple, widely applicable formalism for designing "error-divisible" two qubit gates: a quantum gate set where fractional rotations have proportionally reduced error compared to the full entangling gate. In current noisy intermediate-scale quantum (NISQ) algorithms, performance is largely constrained by error proliferation at high circuit depths, of which two-qubit gate error is generally the dominant contribution. Further, in many hardware implementations, arbitrary two qubit rotations must be composed from multiple two-qubit stock gates, further increasing error. This work introduces a set of criteria, and example waveforms and protocols to satisfy them, using superconducting qubits with tunable couplers for constructing continuous gate sets with significantly reduced error for small-angle rotations. If implemented at scale, NISQ algorithm performance would be significantly improved by our error-divisible gate protocols., Comment: 5 pages main text, 11 pages total, 5 figures main text, 10 figures total, 3 tables in appendix

The rapidly growing quantum information science and engineering (QISE) industry will require both quantum-aware and quantum-proficient engineers at the bachelor's level. We provide a roadmap for building a quantum engineering education program to satisfy this need. For quantum-aware engineers, we describe how to design a first quantum engineering course accessible to all STEM students. For the education and training of quantum-proficient engineers, we detail both a quantum engineering minor accessible to all STEM majors, and a quantum track directly integrated into individual engineering majors. We propose that such programs typically require only three or four newly developed courses that complement existing engineering and science classes available on most larger campuses. We describe a conceptual quantum information science course for implementation at any post-secondary institution, including community colleges and military schools. QISE presents extraordinary opportunities to work towards rectifying issues of inclusivity and equity that continue to be pervasive within engineering. We present a plan to do so and describe how quantum engineering education presents an excellent set of education research opportunities. Finally, we outline a hands-on training plan on quantum hardware, a key component of any quantum engineering program, with a variety of technologies including optics, atoms and ions, cryogenic and solid-state technologies, nanofabrication, and control and readout electronics. Our recommendations provide a flexible framework that can be tailored for academic institutions ranging from teaching and undergraduate-focused two- and four-year colleges to research-intensive universities., Comment: 25 pages, 2 figures

Tomography is an indispensable part of quantum computation as it enables diagnosis of a quantum process through state reconstruction. Existing tomographic protocols are based on determining expectation values of various Pauli operators which typically require single-qubit rotations. However, in realistic systems, qubits often develop some form of unavoidable stray coupling making it difficult to manipulate one qubit independent of its partners. Consequently, standard protocols applied to those systems result in unfaithful reproduction of the true quantum state. We have developed a protocol, called coupling compensated tomography, that can correct for errors due to parasitic couplings completely in software and accurately determine the quantum state. We demonstrate the performance of our scheme on a system of two transmon qubits with always-on $\textit{ZZ}$ coupling. Our technique is a generic tomography tool that can be applied to large systems with different types of stray inter-qubit couplings and facilitates the use of arbitrary tomography pulses and even non-orthogonal axes of rotation., Comment: 7 pages, 4 figures, Supplementary information 9 pages, 6 figures

We consider a range of unconventional modifications to Quantum Annealing (QA), applied to an artificial trial problem with continuously tunable difficulty. In this problem, inspired by "transverse field chaos" in larger systems, classical and quantum methods are steered toward a false local minimum. To go from this local minimum to the global minimum, all N spins must flip, making this problem exponentially difficult to solve. We numerically study this problem by using a variety of new methods from the literature: inhomogeneous driving, adding transverse couplers, and other types of coherent oscillations in the transverse field terms (collectively known as RFQA). We show that all of these methods improve the scaling of the time to solution (relative to the standard uniform sweep evolution) in at least some regimes. Comparison of these methods could help identify promising paths towards a demonstrable quantum speedup over classical algorithms in solving some realistic problems with near-term quantum annealing hardware.

Coherence times for superconducting qubits have greatly improved over time. Moreover, small logical qubit architectures using engineered dissipation have shown great promise for further improvements in the coherence of a logical qubit manifold comprised of few physical qubits. Nevertheless, optimal working parameters for small logical qubits are generally not well understood. This work presents several approaches to finding preferential parameter configurations by looking at three different cases of increasing complexity. We begin by looking at state stabilization of a single qubit using dissipation via coupling to a lossy object. We look at the limiting factors in this approach to error correction, and how we address those by numerically optimizing the parametric coupling strength with the lossy object having an effective time-varying dissipation rate---we call this a pulse-reset cycle. We then translate this approach to more efficient state stabilization to an abstracted three-qubit flip code, and end by looking at the Very Small Logical Qubit (VSLQ). By using these techniques, we can further increase logical state lifetimes for different architectures. We show significant advantages in using a pulse-reset cycle over numerically optimized, fixed parameter spaces., Comment: 10 pages, 8 figures, 1 table

The Markov decision process is the mathematical formalization underlying the modern field of reinforcement learning when transition and reward functions are unknown. We derive a pseudo-Boolean cost function that is equivalent to a K-spin Hamiltonian representation of the discrete, finite, discounted Markov decision process with infinite horizon. This K-spin Hamiltonian furnishes a starting point from which to solve for an optimal policy using heuristic quantum algorithms such as adiabatic quantum annealing and the quantum approximate optimization algorithm on near-term quantum hardware. In proving that the variational minimization of our Hamiltonian is equivalent to the Bellman optimality condition we establish an interesting analogy with classical field theory. Along with proof-of-concept calculations to corroborate our formulation by simulated and quantum annealing against classical Q-Learning, we analyze the scaling of physical resources required to solve our Hamiltonian on quantum hardware.

Using optimal phasor measurement unit placement as a prototypical problem, we assess the computational viability of the current generation D-Wave Systems 2000Q quantum annealer for power systems design problems. We reformulate minimum dominating set for the annealer hardware, solve the reformulation for a standard set of IEEE test systems, and benchmark solution quality and time to solution against the CPLEX Optimizer and simulated annealing. For some problem instances the 2000Q outpaces CPLEX. For instances where the 2000Q underperforms with respect to CPLEX and simulated annealing, we suggest hardware improvements for the next generation of quantum annealers.

The simulation complexity of predicting the time evolution of delocalized many-body quantum systems has attracted much recent interest, and simulations of such systems in real quantum hardware are promising routes to demonstrating a quantum advantage over classical machines. In these proposals, random noise is an obstacle that must be overcome for a faithful simulation, and a single error event can be enough to drive the system to a classically trivial state. We argue that this need not always be the case, and consider a modification to a leading quantum sampling problem-- time evolution in an interacting Bose-Hubbard chain of transmon qubits [Neill et al, Science 2018] -- where each site in the chain has a driven coupling to a lossy resonator and particle number is no longer conserved. The resulting quantum dynamics are complex and highly nontrivial. We argue that this problem is harder to simulate than the isolated chain, and that it can achieve volume-law entanglement even in the strong noise limit, likely persisting up to system sizes beyond the scope of classical simulation. Further, we show that the metrics which suggest classical intractability for the isolated chain point to similar conclusions in the noisy case. These results suggest that quantum sampling problems including nontrivial noise could be good candidates for demonstrating a quantum advantage in near-term hardware., Comment: 20 pages, 15 figures

We theoretically analyze a scheme for fast stabilization of arbitrary qubit states with high fidelities, extending a protocol recently demonstrated experimentally [Lu et al., Phys. Rev. Lett. 119, 150502 (2017)]. That experiment utilized red and blue sideband transitions in a system composed of a fluxonium qubit, a low-Q LC-oscillator, and a coupler enabling us to tune the interaction between them. Under parametric modulations of the coupling strength, the qubit can be steered into any desired pure or mixed single-qubit state. For realistic circuit parameters, we predict that stabilization can be achieved within 100 ns. By varying the ratio between the oscillator's damping rate and the effective qubit-oscillator coupling strength, we can switch between under-damped, critically-damped, and over-damped stabilization and find optimal working points. We further analyze the effect of thermal fluctuations and show that the stabilization scheme remains robust for realistic temperatures.

Quantum annealing is a powerful alternative model for quantum computing, which can succeed in the presence of environmental noise even without error correction. However, despite great effort, no conclusive proof of a quantum speedup (relative to state of the art classical algorithms) has been shown for these systems, and rigorous theoretical proofs of a quantum advantage generally rely on exponential precision in at least some aspects of the system, an unphysical resource guaranteed to be scrambled by random noise. In this work, we propose a new variant of quantum annealing, called RFQA, which can maintain a scalable quantum speedup in the face of noise and modest control precision. Specifically, we consider a modification of flux qubit-based quantum annealing which includes random, but coherent, low-frequency oscillations in the directions of the transverse field terms as the system evolves. We show that this method produces a quantum speedup for finding ground states in the Grover problem and quantum random energy model, and thus should be widely applicable to other hard optimization problems which can be formulated as quantum spin glasses. Further, we show that this speedup should be resilient to two realistic noise channels ($1/f$-like local potential fluctuations and local heating from interaction with a finite temperature bath), and that another noise channel, bath-assisted quantum phase transitions, actually accelerates the algorithm and may outweigh the negative effects of the others. The modifications we consider have a straightforward experimental implementation and could be explored with current technology., Comment: 21 pages, 7 figures

In a recent remarkable experiment [P. Roushan et al., Nature Physics 13, 146 (2017)], a spin current in an architecture of three superconducting qubits was produced during a few microseconds by creating synthetic magnetic fields. The life-time of the current was set by the typical dissipative mechanisms that occur in those systems. We propose a scheme for the generation of permanent currents, even in the presence of such imperfections, and scalable to larger system sizes. It relies on striking a subtle balance between multiple nonequilibrium drives and the dissipation mechanisms, in order to engineer and stimulate chiral excited states which can carry current., Comment: 4 pages, 3 figures

Historically, noise in superconducting circuits has been considered an obstacle to be removed. A large fraction of the research effort in designing superconducting circuits has focused on noise reduction, with great success, as coherence times have increased by four orders of magnitude in the past two decades. However, noise and dissipation can never be fully eliminated, and further, a rapidly growing body of theoretical and experimental work has shown that carefully tuned noise, in the form of engineered dissipation, can be a profoundly useful tool in designing and operating quantum circuits. In this article, I review important applications of engineered dissipation, including state generation, state stabilization, and autonomous quantum error correction, where engineered dissipation can mitigate the effect of intrinsic noise, reducing logical error rates in quantum information processing. Further, I provide a pedagogical review of the basic noise processes in superconducting qubits (photon loss and phase noise), and argue that any dissipative mechanism which can correct photon loss errors is very likely to automatically suppress dephasing. I also discuss applications for quantum simulation, and possible future research directions., Comment: 24 pages, 12 figures; topical review article

We autonomously stabilize arbitrary states of a qubit through parametric modulation of the coupling between a fixed frequency qubit and resonator. The coupling modulation is achieved with a tunable coupler design, in which the qubit and the resonator are connected in parallel to a superconducting quantum interference device. This allows for quasi-static tuning of the qubit-cavity coupling strength from 12 MHz to more than 300 MHz. Additionally, the coupling can be dynamically modulated, allowing for single photon exchange in 6 ns. Qubit coherence times exceeding 20 $\mu$s are maintained over the majority of the range of tuning, limited primarily by the Purcell effect. The parametric stabilization technique realized using the tunable coupler involves engineering the qubit bath through a combination of photon non-conserving sideband interactions realized by flux modulation, and direct qubit Rabi driving. We demonstrate that the qubit can be stabilized to arbitrary states on the Bloch sphere with a worst-case fidelity exceeding 80 %.

This comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power law interactions, $r^{-\alpha}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has been earlier argued [1, 2] that this model obeys the dimensional constraint suggesting the delocalization of all finite temperature states in thermodynamic limit for $\alpha \leq 2d$ in a $d$-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model in Ref. [3]. To resolve this controversy we reexamine the model of Ref. [3] and demonstrate that the infinite temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization [2] is extended to small exponents $0 \leq \alpha \leq d$. Disagreements between two works are explained by the non-standard selection of investigated states in the ordered phase and misinterpretation of the localization-delocalization transition in Ref. [3]., Comment: To be submitted to Physical Review A

One of the largest obstacles to building a quantum computer is gate error, where the physical evolution of the state of a qubit or group of qubits during a gate operation does not match the intended unitary transformation. Gate error stems from a combination of control errors and random single qubit errors from interaction with the environment. While great strides have been made in mitigating control errors, intrinsic qubit error remains a serious problem that sets the primary limit for gate fidelity in modern superconducting qubit architectures. Simultaneously, recent developments of small error-corrected logical qubit devices promise significant increases in logical state lifetime, but translating those improvements into increases in gate fidelity is a complex challenge. In this Letter, we propose a new formalism for implementing gates on and between small logical qubit devices which inherit the parent device's tolerance to single qubit errors which occur at any time before or during the gate. Using a standard phenomenological noise model for superconducting qubits, we demonstrate a universal one- and two-qubit gate set with error rates an order of magnitude lower than those for equivalent operations on single qubits or pairs of qubits, running for the same total duration. The effective logical gate error rate in these models displays superlinear error reduction with linear increases in single qubit lifetime, proving that passive error correction is capable of increasing gate fidelity. These developments further suggest that incorporating small logical qubits into a measurement based code could substantially improve code performance., Comment: 8 pages, 5 figures, includes supplemental information

Superconducting qubits are among the most promising platforms for building a quantum computer. However, individual qubit coherence times are not far past the scalability threshold for quantum error correction, meaning that millions of physical devices would be required to construct a useful quantum computer. Consequently, further increases in coherence time are very desirable. In this letter, we blueprint a simple circuit consisting of two transmon qubits and two additional lossy qubits or resonators, which is passively protected against all single qubit quantum error channels through a combination of continuous driving and engineered dissipation. Photon losses are rapidly corrected through two-photon drive fields implemented with driven SQUID couplings, and dephasing from random potential fluctuations is heavily suppressed by the drive fields used to implement the multi-qubit Hamiltonian. Comparing our theoretical model to published noise estimates from recent experiments on flux and transmon qubits, we find that logical state coherence could be improved by a factor of forty or more compared to the individual qubit $T_1$ and $T_2$ using this technique. We thus demonstrate that there is substantial headroom for improving the coherence of modern superconducting qubits with a fairly modest increase in device complexity.

By coupling pairs of superconducting qubits through a small Josephson junction with a time-dependent flux bias, we show that arbitrary interactions involving any combination of Pauli matrices can be generated with a small number of drive tones applied through the flux bias of the coupling junction. We then demonstrate that similar (though not fully universal) results can be achieved in capacitively coupled qubits by exploiting the higher energy states of the devices through multi-photon drive signals applied to the qubits' flux degrees of freedom. By using this mechanism to couple a qubit to a detuned resonator, the qubit's rotating frame state can be non-destructively measured along any direction on the Bloch sphere. Finally, we describe how the frequency-converting nature of the couplings can be used to engineer a mechanism analogous to dynamic nuclear polarization in NMR systems, capable of cooling an array of qubits well below the ambient temperature, and outline how higher order interactions, such as local 3-body terms, can be engineered through the same couplings. Our results demonstrate that a programmable quantum simulator for large classes of interacting spin models could be engineered with the same physical hardware., Comment: 8 pages, 1 figure

A physical realization of self correcting quantum code would be profoundly useful for constructing a quantum computer. In this theoretical work, we provide a partial solution to major challenges preventing self correcting quantum code from being engineered in realistic devices. We consider a variant of Kitaev's toric code coupled to propagating bosons, which induce a long-ranged interaction between anyonic defects. By coupling the primary quantum system to an engineered dissipation source through resonant energy transfer, we demonstrate a "rate barrier" which leads to a potentially enormous increase in the system's quantum state lifetime through purely passive quantum error correction, even when coupled to an infinite temperature bath. While our mechanism is not scalable to infinitely large systems, the maximum effective size can be very large, and it is fully compatible with active error correction schemes. Our model uses only on-site and nearest-neighbor interactions, and could be implemented in superconducting qubits. We sketch one such implementation at the end of this work., Comment: 19 pages (with references), 12 figures

Recent progress in nanoscale quantum optics and superconducting qubits has made the creation of strongly correlated, and even topologically ordered, states of photons a real possibility. Many of these states are gapped and exhibit anyon excitations, which could be used for a robust form of quantum information processing. However, while numerous qubit array proposals exist to engineer the Hamiltonian for these systems, the question of how to stabilize the many-body ground state of these photonic quantum simulators against photon losses remains largely unanswered. We here propose a simple mechanism which achieves this goal for abelian and non-abelian fractional quantum Hall states of light. Our construction uses a uniform two-photon drive field to couple the qubits of the primary lattice with an auxiliary ``shadow" lattice, composed of qubits with a much faster loss rate than the qubits of the primary quantum simulator itself. This coupling causes hole states created by photon losses to be rapidly refilled, and the system's many-body gap prevents further photons from being added once the strongly correlated ground state is reached. The fractional quantum Hall state (with a small, transient population of quasihole excitations) is thus the most stable state of the system, and all other configurations will relax toward it over time. The physics described here could be implemented in a circuit QED architecture, and the device parameters needed for our scheme to succeed are in reach of current technology. We also propose a simple 6 qubit device, which could easily be built in the near future, that can act as a proof of principle for our scheme., Comment: 9 pages, 6 figures

The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases., Comment: 12 pages, 7 figures. As published

We describe a method for engineering local $k+1$-body interactions ($k=1,2,3$) from two-body couplings in spin-${1}{2}$ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit abelian and non-abelian anyon excitations. The most complex of these, with $k=3$, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model up to small corrections, and further, we describe the process by which one might create and manipulate non-abelian vortices in these circuits, allowing for direct control of the system's quantum information content., Comment: 4 pages + references and supplemental information

We describe a lattice of asymmetrical qubit pairs in one or two dimensions, with couplings arranged so that the motion of single-qubit excited states mimics the behavior of charged lattice bosons hopping in a magnetic field. We show in particular that one can choose the parameters of the many-body circuit to reach a regime where the complex hopping phase between any two elements can be tuned to any value by simply adjusting the relative phases of two applied oscillating voltage signals. We also propose a specific realization of our model using coupled three junction flux qubits, in which one can reach the strongly interacting bosonic quantum Hall limit where one will find anyonic excitations. The circuits could be used for topological quantum computation., Comment: 9 pages (with references), 5 figures, accepted in Phys. Rev. A

We report on a numerical experiment in which we use time-dependent potentials to braid non-abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where $\nu$, the ratio of particles to flux quanta, is near 1/2, 1 or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for $\nu$ near $1$ and $3/2$, with Berry matrices respectively consistent with Ising and Fibonacci anyons. Near $\nu=1/2$, the braids commute., Comment: 5 pages, 1 figure

The electron-electron scattering rate of low-energy quasiparticles is computed perturbatively for a two-dimensional metal with a partially nested Fermi surface, a weak electron-electron interaction and an energy-independent impurity scattering rate. We show that even when the nested section comprises only a small fraction of the Fermi surface, the low energy quasiparticle decay rates throughout the Fermi surface gain a term which is linear in energy $\omega$. The coefficient of this linear contribution is largest along the nested regions but remains finite elsewhere, and is strongly dependent on the impurity scattering rate $\delta$. We discuss the possible relevance of this result to unconventional superconductors (such as the cuprates) and to the normal state of charge density wave materials such as the rare Earth tellurides., Comment: This paper has been withdrawn due to an error in equation (9), which invalidates the subsequent calculations. A revised version is in preparation. Contact: ek359@cornell.edu

We show how interpenetrating optical lattices containing Bose-Fermi mixtures can be constructed to emulate the thermodynamics of quantum electrodynamics (QED). We present models of neutral atoms on lattices in 1+1, 2+1 and 3+1 dimensions whose low energy effective action reduces to that of photons coupled to Dirac fermions of the corresponding dimensionality. We give special attention to 2+1 dimensional electrodynamics (QED3) and discuss how two of its most interesting features, chiral symmetry breaking and Chern-Simons physics, could be observed experimentally., Comment: 11 pages, 5 figures, submitted to Phys. Rev. A

We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wavefunctions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wavefunction is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics., Comment: 4 pages, 3 figures. Published version

We study how different many body states appear in a quantum gas microscope, such as the one developed at Harvard [Bakr et al. Nature 462, 74 (2009)], where the site-resolved parity of the atom number is imaged. We calculate the spatial correlations of the microscope images, corresponding to the correlation function of the parity of the number of atoms at each site. We produce analytic results for a number of well-known models: noninteracting bosons, the large U Bose-Hubbard model, and noninteracting fermions. We find that these parity correlations tend to be less strong than density-density correlations, but they carry similar information., Comment: 8 pages, 4 figures. Published version

We present a method for constructing parent Hamiltonians for the chiral spin liquid. We find two distinct Hamiltonians for which the chiral spin liquid on a square lattice is an exact zero-energy ground state. We diagonalize both Hamiltonians numerically for 16-site lattices, and find that the chiral spin liquid, modulo its two-fold topological degeneracy, is indeed the unique ground state for one Hamiltonian, while it is not unique for the other., Comment: 14 pages, 3 figures; references added

The interacting symplectic fermion model in two spatial dimensions is further analyzed. As an effective low energy theory, the model is unitary. We show that a relativistic mass m is dynamically generated and derive a gap equation for it. By incorporating a finite temperature we study some fundamental properties of the model, such as the specific heat and spin response, which clearly show non-Fermi liquid properties. We find that various physical properties are suppressed at temperatures T< T* where the cross-over scale is T* = m. As a simplified, toy model of high Tc superconductivity, we thus identify the pseudogap energy scale with the zero temperature relativistic mass m, and show that this reproduces some qualitative aspects of the observed phenomenology of the pseudogap. The effects of the pseudogap and finite temperature on the d-wave gap equation are analyzed. In this model, the pseudogap is a distinct phenomenon from superconductivity and in fact competes with it. Our analysis of Tc suggests that the quantum critical point of our model, where the pseudogap vanishes, occurs inside the superconducting dome near optimal doping. For an antiferromagnetic exchange energy of J/k_B ~ 1350K, solutions of the d-wave gap equation give a maximum Tc of about 110K., Comment: 31 pages, 13 figures

A free quantum field theory with Lorentz symmetry is derived for spin-half symplectic fermions in 2+1 dimensions. In particular, we show that fermionic spin-half fields may be canonically quantized in a free theory with a Klein-Gordon Lagrangian. This theory is shown to have all the required properties of a consistent free quantum field theory, namely causality, unitarity, adherence to the spin-statistics theorem, CPT symmetry, and the Hermiticity and positive definiteness of the Hamiltonian. The global symmetry of the free theory is Sp(4) $\simeq$ SO(5). Possible interacting theories of both the pseudo-Hermitian and Hermitian variety are then examined briefly., Comment: 11 pages. v2 with extra material

Demanding a consistent quantum field theory description of spin 1/2 particles near a circular Fermi surface in 2d leads to a unique fermionic theory with relevant quartic interactions which has an emergent Lorentz symmetry and automatically has an Sp(4) = SO(5) internal symmetry. The free theory has resistivity linear in temperature. The interacting theory has a low-energy interacting fixed point and is thus a non-Landau/Fermi liquid. Anti-ferromagnetic (AF) and superconducting (SC) order parameters are bilinears in the fields and form the 5-dimensional vector representation of SO(5). An AF phase occurs at low doping which terminates in a first order transition. We incorporate momentum dependent scattering of Cooper pairs near the Fermi surface to 1-loop and derive a new kind of SC gap equation beyond mean field with a d-wave gap solution. Taking into account the renormalization group (RG) scaling properties near the low energy fixed point, we calculate the complete phase diagram as a function of doping, which shows some universal geometric features. The d-wave SC dome terminates on the over-doped side at the fixed point of the RG,which is a quantum critical point. Optimal doping is estimated to occur just below 3/2\pi^2. The critical temperature for SC at optimal doping is set mainly by the universal nodal Fermi velocity and lattice spacing, and is estimated to average around 140K for LSCO. The pseudogap energy scale is identified as the RG scale of the coupling., Comment: published version