We study the generation of two-qudit entangling quantum logic gates using two techniques in quantum optimal control. We take advantage of both continuous, Lie algebraic control and digital, Lie group control. In both cases, the key is access to a time-dependent Hamiltonian, which can generate an arbitrary unitary matrix in the group SU($d^2$). We find efficient protocols for creating high-fidelity entangling gates. As a test of our theory, we study the case of qudits robustly encoded in nuclear spins of alkaline earth atoms and manipulated with magnetic and optical fields, with entangling interactions arising from the well-known Rydberg blockade. We applied this in a case study based on a $d=10$ dimensional qudit encoded in the $I=9/2$ nuclear spin in ${}^{87}\mathrm{Sr}$, controlled through a combination of nuclear spin resonance, a tensor ac-Stark shift, and Rydberg dressing, which allows us to generate an arbitrary symmetric entangling two-qudit gate, such as CPhase. Our techniques can be used to implement qudit entangling gates for any $2\le d\le 10$ encoded in the nuclear spin. We also studied how decoherence due to the finite lifetime of the Rydberg states affects the creation of the CPhase gate and found, through numerical optimization, a fidelity of $0.9985$, $0.9980$, $0.9942$, and $0.9800$ for $d=2$, $d=3$, $d=5$, and $d=7$, respectively. This provides a powerful platform to explore the various applications of quantum information processing of qudits, including metrological enhancement with qudits, quantum simulation, universal quantum computation, and quantum error correction.

• Received 22 December 2022
• Accepted 30 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040333

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Projective measurement, a basic operation in quantum mechanics, can induce seemingly nonlocal effects. In this work, we analyze such effects in many-body systems by studying the nonequilibrium dynamics of weakly monitored quantum circuits, focusing on entanglement generation and information spreading. We find that, due to measurements, the entanglement dynamics in monitored circuits is indeed “faster” than that of unitary ones in several ways. Specifically, we find that a pair of well-separated regions can become entangled in a time scale ${\ell }^{2/3}$, sublinear in their distance $\ell$. For the case of Clifford monitored circuits, this originates from superballistically growing stabilizer generators of the evolving state. In addition, we find initially local information can spread superlinearly as $t^{3/2}$. Furthermore, by viewing the dynamics as a dynamical encoding process, we show that the superlinear growing length scale relates to an encoding time that is sublinear in system size. To quantify the information dynamics, we develop a formalism generalizing operator spreading to nonunitary dynamics, which is of independent interest.

• Received 12 May 2023
• Accepted 31 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040332

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We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design. Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup. We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries. Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group. This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.

• Received 18 July 2023
• Accepted 25 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040331

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Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the so-called pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features. (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom theorem. Namely, the average error probability of sending $M$ messages through a classical-quantum channel is upper bounded by the minimum error of distinguishing the joint channel input-output state against $(M-1)$ decoupled product states. (ii) Our bound directly yields asymptotic achievability results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality or the quantum union bound are no longer needed. Hence, the derived one-shot bound sharpens existing results relying on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable $\epsilon$-one-shot capacity for classical communication over quantum channels heretofore, improving the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot achievability for classical data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols.

• Received 28 April 2023
• Revised 20 September 2023
• Accepted 16 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040330

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Quantum systems are inherently open and susceptible to environmental noise, which can have both detrimental and beneficial effects on their dynamics. This phenomenon has been observed in biomolecular systems, where noise enables novel functionalities, making the simulation of their dynamics a crucial target for digital and analog quantum simulation. Nevertheless, the computational capabilities of current quantum devices are often limited due to their inherent noise. In this work, we present a novel approach that capitalizes on the intrinsic noise of quantum devices to reduce the computational resources required for simulating open quantum systems. Our approach combines quantum noise characterization methods with quantum error mitigation techniques, enabling us to manipulate and control the intrinsic noise in a quantum circuit. Specifically, we selectively enhance or reduce decoherence rates in the quantum circuit to achieve the desired simulation of open-system dynamics. We provide a detailed description of our methods and report on the results of noise characterization and quantum error mitigation experiments conducted on both real and emulated IBM Quantum computers. Additionally, we estimate the experimental resource requirements for our techniques. Our approach holds the potential to unlock new simulation techniques in noisy intermediate-scale quantum devices, harnessing their intrinsic noise to enhance quantum computations.

• Received 15 March 2023
• Accepted 10 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040329

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We provide a new perspective on shadow tomography by demonstrating its deep connections with the general theory of measurement frames. By showing that the formalism of measurement frames offers a natural framework for shadow tomography—in which “classical shadows” correspond to unbiased estimators derived from a suitable dual frame associated with the given measurement—we highlight the intrinsic connection between standard state tomography and shadow tomography. Such a perspective allows us to examine the interplay between measurements, reconstructed observables, and the estimators used to process measurement outcomes, while paving the way to assessing the influence of the input state and the dimension of the underlying space on estimation errors. Our approach generalizes the method described by Huang et al. [H.-Y. Huang et al., Nat. Phys. 16, 1050 (2020)], whose results are recovered in the special case of covariant measurement frames. As an application, we demonstrate that a sought-after target of shadow tomography can be achieved for the entire class of tight rank-1 measurement frames—namely, that it is possible to accurately estimate a finite set of generic rank-1 bounded observables while avoiding the growth of the number of the required samples with the state dimension.

• Received 13 February 2023
• Revised 24 July 2023
• Accepted 2 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040328

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This work explores the possibility of creating and controlling unconventional nonlinearities by periodic driving, in a broad class of systems described by the nonlinear Schrödinger equation (NLSE). By means of a parent quantum many-body description, we demonstrate that such driven systems are well captured by an effective NLSE with emergent nonlinearities, which can be finely controlled by tuning the driving sequence. We first consider a general class of two-mode nonlinear systems—relevant to optical Kerr cavities, waveguides, and Bose-Einstein condensates—where we find an emergent four-wave mixing nonlinearity, which originates from pair-hopping processes in the parent quantum picture. Tuning this drive-induced nonlinearity is shown to modify the phase-space topology, which can be detected through relative population and phase measurements, and also leads to enhanced quantum properties such as spin squeezing. We then couple individual (two-mode) dimers in view of designing extended lattice models with unconventional nonlinearities and controllable pair-hopping processes. Following this general dimerization construction, we obtain an effective lattice model with drive-induced interactions, whose ground state exhibits orbital order, chiral currents, and emergent magnetic fluxes through the spontaneous breaking of time-reversal symmetry. We analyze these intriguing properties both in the weakly interacting (mean-field) regime, captured by the effective NLSE, and in the strongly correlated quantum regime. Our general approach opens a route for the engineering of unconventional optical nonlinearities in photonic devices and controllable drive-induced interactions in ultracold quantum matter.

• Received 15 March 2022
• Revised 19 April 2023
• Accepted 14 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040327

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We introduce a classical algorithm for sampling the output of shallow, noisy random circuits on two-dimensional qubit arrays. The algorithm builds on the recently proposed “space-evolving block decimation” (SEBD) [Napp et al, Phys. Rev. X 12, 021021 (2022)] and extends it to the case of noisy circuits. SEBD is based on a mapping of two-dimensional unitary circuits to one-dimensional monitored ones, which feature measurements alongside unitary gates; it exploits the presence of a measurement-induced entanglement phase transition to achieve efficient (approximate) sampling below a finite critical depth $T_c$. Our noisy-SEBD algorithm unravels the action of noise into measurements, further lowering entanglement and enabling efficient classical sampling up to larger circuit depths. We analyze a class of physically relevant noise models (unital qubit channels) within a two-replica statistical mechanics treatment, finding weak measurements to be the optimal (i.e., most disentangling) unraveling. We then locate the noisy-SEBD complexity transition as a function of circuit depth and noise strength in realistic circuit models. As an illustrative example, we show that circuits on heavy-hexagon qubit arrays with noise rates of approximately equal to $2\mathrm{\%}$ per cnot, based on IBM Quantum processors, can be efficiently sampled up to a depth of five $i$swap (or ten cnot) gate layers. Our results help sharpen the requirements for practical hardness of simulation of noisy hardware.

• Received 18 July 2023
• Accepted 19 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040326

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We study quantum interior-point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity and/or depth of non-Clifford $T$ gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant prefactors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage.

• Received 17 January 2023
• Accepted 25 August 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040325

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Reconstructing the Hamiltonian of a quantum system is an essential task for characterizing and certifying quantum processors and simulators. Existing techniques either rely on projective measurements of the system before and after coherent time evolution and do not explicitly reconstruct the full time-dependent Hamiltonian or interrupt evolution for tomography. Here, we experimentally demonstrate that an a priori unknown, time-dependent Hamiltonian can be reconstructed from continuous weak measurements concurrent with coherent time evolution in a system of two superconducting transmons coupled by a flux-tunable coupler. In contrast to previous work, our technique does not require interruptions, which would distort the recovered Hamiltonian. We introduce an algorithm, which recovers the Hamiltonian and density matrix from an incomplete set of continuous measurements, and demonstrate that it reliably extracts amplitudes of a variety of single-qubit and entangling two-qubit Hamiltonians. We further demonstrate how this technique reveals deviations from a theoretical control Hamiltonian, which would otherwise be missed by conventional techniques. Our work opens up novel applications for continuous weak measurements, such as studying nonidealities in gates, certifying analog quantum simulators, and performing quantum metrology.

• Received 2 February 2023
• Revised 11 September 2023
• Accepted 10 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040324

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$W$ states are a valuable resource for various quantum information tasks, and several protocols to generate them have been proposed and implemented. We introduce a quantum repeater protocol to efficiently distribute three-qubit $W$ states over arbitrary distances in a two-dimensional triangular quantum network with polylogarithmic overhead, thereby enabling these applications between remote parties. The repeater protocol combines two ingredients that we establish: probabilistic entanglement swapping with three copies of three-qubit $W$ states to a single long-distance three-qubit $W$ state, and an improved entanglement purification protocol. The latter not only shows a better performance, but also an enlarged purification regime as compared to previous approaches. We show that the repeater protocol allows one to deal with errors resulting from imperfect channels or state preparation, and noisy operations, and we analyze error thresholds, achievable fidelities, and overheads.

• Received 2 May 2023
• Revised 8 September 2023
• Accepted 4 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040323

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Quantum threshold theorems impose hard limits on the hardware capabilities to process quantum information. We derive tight and fundamental upper bounds to loss-tolerance thresholds in different linear-optical quantum information processing settings through an adversarial framework, taking into account the intrinsically probabilistic nature of linear optical Bell measurements. For logical Bell state measurements—ubiquitous operations in photonic quantum information—we demonstrate analytically that linear optics can achieve the fundamental loss threshold imposed by the no-cloning theorem even though, following the work of Lee et al. [Phys. Rev. A 100, 052303 (2019)] the constraint was widely assumed to be stricter. We spotlight the assumptions of the latter publication and find their bound holds for a logical Bell measurement built from adaptive physical linear-optical Bell measurements. We also give an explicit even stricter bound for nonadaptive Bell measurements.

• Received 21 February 2023
• Revised 22 May 2023
• Accepted 22 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040322

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Tunable couplers have recently become one of the most powerful tools for implementing two-qubit gates between superconducting qubits. A tunable coupler typically includes a nonlinear element, such as a superconducting quantum interference device, which is used to tune the resonance frequency of an $LC$ circuit connecting two qubits. Here we propose a complimentary approach where instead of tuning the resonance frequency of the tunable coupler by applying a quasistatic control signal, we excite by microwave the degree of freedom associated with the coupler itself. Because of strong effective longitudinal coupling between the coupler and the qubits, the frequency of this transition strongly depends on the computational state, leading to different phase accumulations in different states. Using this method, we experimentally demonstrate a controlled-$Z$ gate of 44-ns duration on a fluxonium-based quantum processor, obtaining a fidelity of $97.6\mathrm{\%}\pm 0.4\mathrm{\%}$ characterized by cross-entropy benchmarking.

• Received 22 February 2023
• Accepted 5 October 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040321

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The security of prepare-and-measure satellite-based quantum key distribution (QKD), under restricted eavesdropping scenarios, is addressed. We particularly consider cases where the eavesdropper, Eve, has limited access to the transmitted signal by Alice and/or Bob’s receiver station. This restriction is modeled by lossy channels between relevant parties, where the transmissivity of such channels can, in principle, be bounded by monitoring techniques. An artifact of such lossy channels is the possibility of having bypass channels, those that are not accessible to Eve but that may not necessarily be characterized by the users either. This creates interesting unexplored scenarios for analyzing QKD security. In this paper, we obtain generic bounds on the key rate in the presence of bypass channels and apply them to continuous-variable QKD protocols with Gaussian encoding with direct and reverse reconciliation. We find regimes of operation in which the above restrictions on Eve can considerably improve system performance. We also develop customized bounds for several protocols in the BB84 family and show that, in certain regimes, even the simple protocol of BB84 with weak coherent pulses is able to offer positive key rates at high channel losses, which would otherwise be impossible under an unrestricted Eve. In this case, the limitation on Eve would allow Alice to send signals with larger intensities than the optimal value under an ideal Eve, which effectively reduces the effective channel loss. In all these cases, the part of the transmitted signal that does not reach Eve can play a nontrivial role in specifying the achievable key rate. Our work opens up new security frameworks for spaceborne quantum communications systems.

• Received 20 December 2022
• Revised 28 April 2023
• Accepted 25 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040320

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In the race to build scalable quantum computers, minimizing the resource consumption of their full stack to achieve a target performance becomes crucial. It mandates a synergy of fundamental physics and engineering: the former for the microscopic aspects of computing performance and the latter for the macroscopic resource consumption. For this, we propose a holistic methodology dubbed metric noise resource (MNR) that is able to quantify and optimize all aspects of the full-stack quantum computer, bringing together concepts from quantum physics (e.g., noise on the qubits), quantum information (e.g., computing architecture and type of error correction), and enabling technologies (e.g., cryogenics, control electronics, and wiring). This holistic approach allows us to define and study resource efficiencies as ratios between performance and resource cost. As a proof of concept, we use MNR to minimize the power consumption of a full-stack quantum computer, performing noisy or fault-tolerant computing with a target performance for the task of interest. Comparing this with a classical processor performing the same task, we identify a quantum energy advantage in regimes of parameters distinct from the commonly considered quantum computational advantage. This provides a previously overlooked practical argument for building quantum computers. While our illustration uses highly idealized parameters inspired by superconducting qubits with concatenated error correction, the methodology is universal—it applies to other qubits and error-correcting codes—and it provides experimenters with guidelines to build energy-efficient quantum computers. In some regimes of high energy consumption, it can reduce this consumption by orders of magnitude. Overall, our methodology lays the theoretical foundation for resource-efficient quantum technologies.

• Received 29 November 2022
• Accepted 31 July 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040319

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The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor’s algorithm for factoring products of primes. Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically. While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT. The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits. In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively. Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions. We demonstrate this speedup in test calculations on some simple functions. For data vectors of length ${10}^6$–${10}^8$, the speedup can be a few orders of magnitude.

• Received 2 January 2023
• Accepted 25 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040318

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We introduce a method to measure many-body magic in quantum systems based on a statistical exploration of Pauli strings via Markov chains. We demonstrate that sampling such Pauli-Markov chains gives ample flexibility in terms of partitions where to sample from: in particular, it enables the efficient extraction of the magic contained in the correlations between widely separated subsystems, which characterizes the nonlocality of magic. Our method can be implemented in a variety of situations. We describe an efficient sampling procedure using tree tensor networks, that exploit their hierarchical structure leading to a modest $O(\log N)$ computational scaling with system size. To showcase the applicability and efficiency of our method, we demonstrate the importance of magic in many-body systems via the following discoveries: (a) for one-dimensional systems, we show that long-range magic displays strong signatures of conformal quantum criticality (Ising, Potts, and Gaussian), overcoming the limitations of full state magic; (b) in two-dimensional ${\mathbb{Z}}_2$ lattice gauge theories, we provide conclusive evidence that magic is able to identify the confinement-deconfinement transition, and displays critical scaling behavior even at relatively modest volumes. Finally, we discuss an experimental implementation of the method, which relies only on measurements of Pauli observables.

• Received 13 June 2023
• Accepted 26 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040317

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Bosonic cat qubits stabilized with a driven two-photon dissipation are systems with exponentially biased noise, opening the door to low-overhead, fault-tolerant, and universal quantum computing. However, current gate proposals for such qubits induce substantial noise of the unprotected type, the poor scaling of which with the relevant experimental parameters limits their practical use. In this work, we provide a new perspective on dissipative cat qubits by reconsidering the reservoir mode used to engineer the tailored two-photon dissipation and showing how it can be leveraged to mitigate gate-induced errors. Doing so, we introduce four new designs of high-fidelity and bias-preserving cat-qubit gates and compare them to the prevalent gate methods. These four designs should give a broad overview of gate engineering for dissipative systems with different and complementary ideas. In particular, we propose both already achievable low-error gate designs and longer-term implementations.

• Received 27 March 2023
• Revised 21 July 2023
• Accepted 20 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040316

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Single-qubit sensing protocols can be used to measure qubit-bath coupling parameters. However, for sufficiently large coupling, the sensing protocol itself perturbs the bath, which is predicted to result in a characteristic response in the sensing measurements. Here, we observe this bath perturbation, also known as a quantum quench, by preparing the nuclear spin bath of a nitrogen-vacancy (NV) center in polarized initial states and performing phase-resolved spin-echo measurements on the NV electron spin. These measurements reveal a time-dependent phase determined by the initial state of the bath. We derive the relationship between the sensor phase and the Gaussian spin-bath polarization and apply it to reconstruct both the axial and transverse polarization components. Using this insight, we optimize the transfer efficiency of our dynamic nuclear polarization sequence. This technique for directly measuring bath polarization may assist in preparing high-fidelity quantum memory states, improving nanoscale NMR methods, and investigating non-Gaussian quantum baths.

• Received 3 March 2023
• Accepted 25 September 2023

DOI:https://doi.org/10.1103/PRXQuantum.4.040315

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