Bayes’ rule, $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$, is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes’ rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes’ rules appearing in the literature are all instances of our definition. Moreover, our Bayes’ rule is based upon a simple relationship between the notions of state over time and a time-reversal-symmetry map, both of which are introduced here.

We study multiparameter sensing of two-dimensional (2D) and three-dimensional (3D) vector fields within the Bayesian framework for $\mathrm{SU(2)}$ quantum interferometry. We establish a method to determine the optimal quantum sensor, which establishes the fundamental limit on the precision of simultaneously estimating multiple parameters with an $N$-atom sensor. Keeping current experimental platforms in mind, we present sensors that have limited entanglement capabilities and yet significantly outperform sensors that operate without entanglement and approach the optimal quantum sensor in terms of performance. Furthermore, we show how these sensors can be implemented on current programmable quantum sensors with variational quantum circuits by minimizing a metrological cost function. The resulting circuits prepare tailored entangled states and perform measurements in an appropriate entangled basis to realize the best-possible quantum sensor given the native entangling resources available on a given sensor platform. Notable examples include a 2D and 3D quantum “compass” and a 2D sensor that provides a scalable improvement over unentangled sensors. Our results on optimal and variational multiparameter quantum metrology are useful for advancing precision measurements in fundamental science and ensuring the stability of quantum computers, which can be achieved through the incorporation of optimal quantum sensors in a quantum feedback loop.

In this work, we study the single-shot performance of higher-dimensional hypergraph product codes decoded using belief propagation and ordered-statistics decoding [P. Panteleev and G. Kalachev, Quantum 5, 585 (2021)]. We find that decoding data-qubit and syndrome-measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes. For the three-dimensional toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for $Z$ errors, compared to the threshold of 2.90% previously found using a two-stage decoder [Quintavalle et al., 2021]. For the four-dimensional (4D) toric code, for which both $X$ and $Z$ error correction is single shot, our results are consistent with a sustainable single-shot threshold of 4.3%, which is even higher than the threshold of 2.93% for the two-dimensional toric code for the same noise model but using $L$ rounds of stabilizer measurement. We also explore the performance of balanced-product and 4D hypergraph-product codes, which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.

We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is $\delta /ϵ$, where $ϵ$ is the target precision, and the preconstant $\delta$ can be arbitrarily close to $0$ as the initial state approaches the target eigenstate. The total cost of the algorithm satisfies the Heisenberg-limited scaling $\tilde{\mathcal{O}}({ϵ}^{-1})$. As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers. The key technique is a simple subroutine called quantum complex exponential least squares (QCELS). Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large. If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to $ϵ$. The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps. We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.

We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices. The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions. We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species. Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory. This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that—while pivotal for a successful implementation—have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics. We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope ${}^{173}$$\mathrm{Yb}$, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one- and two-dimensional settings.

The dark-matter puzzle is one of the most important open problems in modern physics. The ultralight axion is a well-motivated dark-matter candidate, conceived to resolve the strong-$CP$ problem of quantum chromodynamics. Numerous precision experiments are searching for the three nongravitational interactions of axionlike dark matter. Some of the searches are approaching fundamental quantum limits on their sensitivity. This Perspective describes several approaches that use quantum engineering to circumvent these limits. Squeezing and single-photon counting can enhance searches for the axion-photon interaction. Optimization of quantum spin-ensemble properties is needed to realize the full potential of spin-based searches for the electric dipole moment and the gradient interactions of axion dark matter. Several metrological and sensing techniques, developed in the field of quantum information science, are finding natural applications in this area of experimental fundamental physics.

We propose a scheme to implement Kitaev’s honeycomb model with cold atoms, based on a periodic (Floquet) drive, in view of realizing and probing non-Abelian chiral spin liquids using quantum simulators. We derive the effective Hamiltonian to leading order in the inverse-frequency expansion, and show that the drive opens up a topological gap in the spectrum without mixing the effective Majorana and vortex degrees of freedom. We address the challenge of probing the physics of Majorana fermions, while having access only to the original composite spin degrees of freedom. Specifically, we propose to detect the properties of the chiral spin liquid phase using gap spectroscopy and edge quenches in the presence of the Floquet drive. The resulting chiral edge signal, which relates to the thermal Hall effect associated with neutral Majorana currents, is found to be robust for realistically prepared states. By combining strong interactions with Floquet engineering, our work paves the way for future studies of non-Abelian excitations and quantized thermal transport using quantum simulators.

Graph codes play an important role in photonic quantum technologies as they provide significant protection against qubit loss, a dominant noise mechanism. Here, we develop methods to analyze and optimize measurement-based tolerance to qubit loss and computational errors for arbitrary graph codes. Using these tools we identify optimized codes with up to 12 qubits and asymptotically large modular constructions. The developed methods enable significant benefits for various photonic quantum technologies, as we illustrate with novel all-photonic quantum repeater states for quantum communication and high-threshold fusion-based schemes for fault-tolerant quantum computing.

We develop a theoretical framework for $S_n$-equivariant convolutional quantum circuits with $\mathrm{SU}(d)$ symmetry, building on and significantly generalizing Jordan’s permutational quantum computing formalism based on Schur-Weyl duality connecting both $\mathrm{SU}(d)$ and $S_n$ actions on qudits. In particular, we utilize the Okounkov-Vershik approach to prove Harrow’s statement on the equivalence between $\mathrm{SU}(d)$ and $S_n$ irrep bases and to establish the $S_n$-equivariant convolutional quantum alternating ansätze ($S_n$-CQA) using Young-Jucys-Murphy elements. We prove that $S_n$-CQA is able to generate any unitary in any given $S_n$ irrep sector, which may serve as a universal model for a wide array of quantum machine-learning problems with the presence of $\mathrm{SU}(d)$ symmetry. Our method provides another way to prove the universality of the quantum approximate optimization algorithm and verifies that four-local $\mathrm{SU}(d)$-symmetric unitaries are sufficient to build generic $\mathrm{SU}(d)$-symmetric quantum circuits up to relative phase factors. We present numerical simulations to showcase the effectiveness of the ansätze to find the ground-state energy of the $J_1$-$J_2$ antiferromagnetic Heisenberg model on the rectangular and kagome lattices. Our work provides the first application of the celebrated Okounkov-Vershik $S_n$ representation theory to quantum physics and machine learning, from which to propose quantum variational ansätze that strongly suggests to be classically intractable tailored towards a specific optimization problem.

Cavity quantum electrodynamics (QED) with cooperativity far greater than unity enables high-fidelity quantum sensing and information processing. The high-cooperativity regime is often reached through the use of short single-mode resonators. More complicated multimode resonators, such as the near-confocal optical Fabry-Prot cavity, can provide intracavity atomic imaging in addition to high cooperativity. This capability has recently proved important for exploring quantum many-body physics in the driven-dissipative setting. In this work, we show that a confocal-cavity–QED microscope can realize cooperativity in excess of 110. This cooperativity is on par with the very best single-mode cavities (which are far shorter) and 21 times greater than single-mode resonators of similar length and mirror radii. The 1.7-$\text{μ}\mathrm{m}$ imaging resolution is naturally identical to the photon-mediated interaction range. We measure these quantities by determining the threshold of cavity superradiance when small optically tweezed Bose-Einstein condensates are pumped at various intracavity locations. Transmission measurements of an ex situ cavity corroborate these results. We provide a theoretical description that shows how cooperativity enhancement arises from the dispersive coupling to the atoms of many near-degenerate modes.

The generalized Bloch decomposition of a bipartite quantum state gives rise to a correlation matrix whose singular values provide rich information about nonlocal properties of the state, such as the dimensionality of entanglement. While some entanglement criteria based on the singular values exist, a complete understanding of the geometry of admissible correlation matrices is lacking. We provide a deeper insight into the geometry of the singular values of the correlation matrices of limited Schmidt number. First, we provide a link to the framework of randomized measurements and show how to obtain knowledge about the singular values in this framework by constructing observables that yield the same moments as one obtains from orthogonal averages over the Bloch sphere. We then focus on the case of separable states and characterize the boundary of the set of the first two nonvanishing moments by giving explicit constructions for some of the faces and extremal points. These constructions yield a connection between the geometry of the correlation matrices and the existence problems of maximal sets of mutually unbiased bases, as well as symmetric, informationally complete, positive operator-valued measures.

We consider the problem of detecting the dimensionality of entanglement with the use of correlations between measurements in randomized directions. First, exploiting the recently derived covariance matrix criterion for the entanglement dimensionality [S. Liu et al., arXiv:2208.04909], we derive an inequality that resembles well-known entanglement criteria, but contains different bounds for the different dimensionalities of entanglement. This criterion is invariant under local changes of $\mathrm{su}(d)$ bases and can be used to find regions in the space of moments of randomized correlations, generalizing the results of [S. Imai et al., Phys. Rev. Lett. 126, 150501 (2021)] to the case of entanglement-dimensionality detection. In particular, we find analytical boundary curves for the different entanglement dimensionalities in the space of second- and fourth-order moments of randomized correlations for all dimensions $d_a=d_b=d$ of a bipartite system. We then show how our method works in practice, also considering a finite statistical sample of correlations, and we also show that it can detect more states than other entanglement-dimensionality criteria available in the literature, thus providing a method that is both very powerful and potentially simpler in practical scenarios. We conclude by discussing the partly open problem of the implementation of our method in the multipartite scenario.

Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner. However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques. We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms. We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law. Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques. We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank. Combining with a tighter error analysis, we show that it suffices to use $({\eta }^{1/3}{n}^{1/3}+n^{2/3}/{\eta }^{2/3})n^{1+o(1)}$ gates to simulate uniform electron gas with $n$ spin orbitals and $\eta$ electrons in second quantization in real space, asymptotically improving over the best previous work. We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation. We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic $n$-qubit two-local Hamiltonians with commuting terms require at least $\mathrm{\Omega }(n^2)$ gates to evolve with accuracy $ϵ=\mathrm{\Omega }(1/\mathrm{poly}(n))$ for time $t=\mathrm{\Omega }(ϵ)$. Our proof is based on a gate-efficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-$2$ subspace, which may be of independent interest. Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.

Using the platform of a trapped-atom clock on a chip, we observe the time evolution of spin-squeezed hyperfine clock states in ultracold rubidium atoms on previously inaccessible time scales up to 1 s. The spin degree of freedom remains squeezed after 0.6 s, which is consistent with the limit imposed by particle loss and is compatible with typical Ramsey times in state-of-the-art microwave clocks. The results also reveal a surprising spin-exchange interaction effect that amplifies the cavity-based spin measurement via a correlation between spin and external degrees of freedom. These results open up perspectives for squeezing-enhanced atomic clocks in a metrologically relevant regime and highlight the importance of spin interactions in real-life applications of spin squeezing.

In efforts to scale the size of quantum computers, modularity plays a central role across most quantum computing technologies. In the light of fault tolerance, this necessitates designing quantum error-correcting codes that are compatible with the connectivity arising from the architectural layouts. In this paper, we aim to bridge this gap by giving a novel way to view and construct quantum low-density parity-check (LDPC) codes tailored for modular architectures. We demonstrate that if the intra- and intermodular qubit connectivity can be viewed as corresponding to some classical or quantum LDPC codes then their hypergraph product code fully respects the architectural connectivity constraints. Finally, we show that relaxed connectivity constraints that allow twists of connections between modules pave a way to construct codes with better parameters.

Oblivious transfer (OT) is an important cryptographic primitive. Any multiparty computation can be realized with OT as building block. xor oblivious transfer (XOT) is a variant where the sender Alice has two bits and a receiver Bob obtains either the first bit, the second bit, or their xor. Bob should not learn anything more than this and Alice should not learn what Bob has learnt. Perfect quantum OT with information-theoretic security is known to be impossible. We determine the smallest possible cheating probabilities for unrestricted dishonest parties in noninteractive quantum XOT protocols using symmetric pure states and present an optimal protocol, which outperforms classical protocols. We also “reverse” this protocol, so that Bob becomes sender of a quantum state and Alice the receiver who measures it, while still implementing oblivious transfer from Alice to Bob. Cheating probabilities for both parties stay the same as for the unreversed protocol. We optically implement both the unreversed and the reversed protocols, and cheating strategies, noting that the reversed protocol is easier to implement.

It is for the first time that quantum simulation for high-energy physics (HEP) is studied in the U.S. decadal particle-physics community planning, and in fact until recently, this was not considered a mainstream topic in the community. This fact speaks of a remarkable rate of growth of this subfield over the past few years, stimulated by the impressive advancements in quantum information sciences (QIS) and associated technologies over the past decade, and the significant investment in this area by the government and private sectors in the U.S. and other countries. High-energy physicists have quickly identified problems of importance to our understanding of nature at the most fundamental level, from tiniest distances to cosmological extents, that are intractable with classical computers but may benefit from quantum advantage. They have initiated, and continue to carry out, a vigorous program in theory, algorithm, and hardware co-design for simulations of relevance to the HEP mission. This Roadmap is an attempt to bring this exciting and yet challenging area of research to the spotlight, and to elaborate on what the promises, requirements, challenges, and potential solutions are over the next decade and beyond.

The pair-coherent state (PCS) is a theoretical extension of the Glauber coherent state to two harmonic oscillators. It is an interesting class of non-Gaussian continuous-variable entangled state and it is also at the heart of a promising quantum error-correction code: the pair-cat code. Here, we report an experimental demonstration of the pair-coherent state of microwave photons in two superconducting cavities. We implement a cross-cavity pair-photon driven-dissipation process, which conserves the photon-number difference between cavities and stabilizes the state to a specific complex amplitude. We further introduce a technique of quantum subspace tomography, which enables direct measurements of individual coherence elements of a high-dimensional quantum state without global tomographic reconstruction. We characterize our two-mode quantum state with up to four photons in each cavity using this subspace tomography together with direct measurements of the photon-number difference and the joint Wigner function. We identify the spurious cross-Kerr interaction between the cavities and our dissipative reservoir mode as a prominent dephasing channel that limits the steady-state coherence in our current scheme. Our experiment provides a set of reservoir-engineering and state-characterization tools to study quantum optics and implement multimode bosonic codes in superconducting circuits.

Quantum simulators have recently enabled experimental observations of the internal thermalization of quantum many-body systems. Often, the global energy and particle number are conserved and the system is prepared with a well-defined particle number—in a microcanonical subspace. However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other. Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information. Until now, this subfield has remained theoretical. We initiate the experimental testing of its predictions, with a trapped-ion simulator. We prepare 6–21 spins in an approximate microcanonical subspace, a generalization of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously. We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations. The noncommuting charges are the three spin components. We find that small subsystems equilibrate to near a recently predicted non-Abelian thermal state. This work bridges quantum many-body simulators to the quantum thermodynamics of noncommuting charges, the predictions of which can now be tested.

Linear strings of trapped atomic ions held in radio-frequency (rf) traps constitute one of the leading platforms for quantum simulation experiments, allowing the investigation of interacting quantum matter. However, linear ion strings have drawbacks, such as the difficulty of scaling beyond approximately $50$ particles as well as the inability to naturally implement spin models with more than one spatial dimension. Here we present experiments with planar Coulomb crystals of up to 105 ${}^{40}{\mathrm{Ca}}^{+}$ ions in a novel monolithic rf trap, laying the groundwork for quantum simulations of two-dimensional spin models with single-particle control. We characterize the trapping potential by analysis of crystal images and compare the observed crystal configurations with numerical simulations. We further demonstrate stable confinement of large crystals, free of structural configuration changes, and find that rf heating of the crystal is not an obstacle for future quantum simulation experiments. Finally, we prepare the out-of-plane motional modes of planar crystals consisting of up to 105 ions close to their ground state by electromagnetically induced transparency cooling, an important prerequisite for implementing long-range entangling interactions.