01595nas a2200145 4500008004100000245005500041210005500096260001300151520116600164100002301330700002001353700002201373700001701395856003701412 2024 eng d00aTopology and entanglement of molecular phase space0 aTopology and entanglement of molecular phase space c3/7/20243 a

We formulate a quantum phase space for molecular rotational and nuclear-spin states. Taking in molecular geometry and nuclear-spin data, our framework yields admissible position and momentum states, inter-convertible via a generalized Fourier transform. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We identify two features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, many molecular species are intrinsically rotation-spin entangled in a way that cannot be broken without transitioning to another species or breaking symmetry. Second, each molecular position state houses an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline scenarios where these features can be experimentally probed.

1 aAlbert, Victor, V.1 aKubischta, Eric1 aLemeshko, Mikhail1 aLiu, Lee, R. uhttps://arxiv.org/abs/2403.0457201560nas a2200157 4500008004100000245005000041210005000091260001400141520111200155100001801267700001601285700001601301700002501317700002301342856003701365 2023 eng d00aBounds on Autonomous Quantum Error Correction0 aBounds on Autonomous Quantum Error Correction c8/30/20233 a

Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an "always-on'' decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. For many-body quantum codes, we show that, to achieve error suppression comparable to active error correction, autonomous decoders generally require correction rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.

1 aShtanko, Oles1 aLiu, Yu-Jie1 aLieu, Simon1 aGorshkov, Alexey, V.1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2308.1623301707nas a2200133 4500008004100000245007300041210006900114260001500183520128600198100001401484700001501498700002301513856003701536 2023 eng d00aClifford operations and homological codes for rotors and oscillators0 aClifford operations and homological codes for rotors and oscilla c11/13/20233 a

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. By interpreting rotor wavefunctions as periodically identified wavefunctions of a harmonic oscillator, we determine the group of bosonic Gaussian operations inherited by the rotor. This n-rotor Clifford group, U(1)n(n+1)/2⋊GLn(Z), is represented by continuous U(1) gates generated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) momentum sign-flip and sum gates. We classify homological rotor error-correcting codes [arXiv:2303.13723] and various rotor states based on equivalence under Clifford operations.
Reversing direction, we map homological rotor codes and rotor Clifford operations back into oscillators by interpreting occupation-number states as rotor states of non-negative angular momentum. This yields new multimode homological bosonic codes protecting against dephasing and changes in occupation number, along with their corresponding encoding and decoding circuits. In particular, we show how to non-destructively measure the oscillator phase using conditional occupation-number addition and post selection. We also outline several rotor and oscillator varieties of the GKP-stabilizer codes [arXiv:1903.12615].

1 aXu, Yijia1 aWang, Yixu1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2311.0767901472nas a2200133 4500008004100000245008700041210006900128260001500197520101000212100002601222700003001248700002301278856003701301 2023 eng d00aNon-invertible symmetry-protected topological order in a group-based cluster state0 aNoninvertible symmetryprotected topological order in a groupbase c12/14/20233 a

Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a G×Rep(G)-symmetric state: the G cluster state introduced in Brell, New Journal of Physics 17, 023029 (2015) [at this http URL]. We show that this state lies in a symmetry-protected topological (SPT) phase protected by G×Rep(G) symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely protected edge modes, string order parameters, and topological response. We discuss how G cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where G is a semidirect product of abelian groups.

1 aFechisin, Christopher1 aTantivasadakarn, Nathanan1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2312.0927201025nas a2200145 4500008004100000245002800041210002800069260001400097520064400111100002200755700002200777700002000799700002300819856003700842 2023 eng d00aQuantum spherical codes0 aQuantum spherical codes c12/7/20233 a

We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs. We also recast concatenations of CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise

1 aJain, Shubham, P.1 aIosue, Joseph, T.1 aBarg, Alexander1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2302.1159301765nas a2200169 4500008004100000245008000041210006900121260001400190300001100204490000600215520126900221100001401490700001501504700001601519700002301535856003701558 2023 eng d00aQubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison0 aQubitOscillator Concatenated Codes Decoding Formalism and Code C c6/14/2023 a0203420 v43 a

Concatenating bosonic error-correcting codes with qubit codes can substantially boost the error-correcting power of the original qubit codes. It is not clear how to concatenate optimally, given that there are several bosonic codes and concatenation schemes to choose from, including the recently discovered Gottesman-Kitaev-Preskill (GKP) – stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)] that allow protection of a logical bosonic mode from fluctuations of the conjugate variables of the mode. We develop efficient maximum-likelihood decoders for and analyze the performance of three different concatenations of codes taken from the following set: qubit stabilizer codes, analog or Gaussian stabilizer codes, GKP codes, and GKP-stabilizer codes. We benchmark decoder performance against additive Gaussian white noise, corroborating our numerics with analytical calculations. We observe that the concatenation involving GKP-stabilizer codes outperforms the more conventional concatenation of a qubit stabilizer code with a GKP code in some cases. We also propose a GKP-stabilizer code that suppresses fluctuations in both conjugate variables without extra quadrature squeezing and formulate qudit versions of GKP-stabilizer codes.

1 aXu, Yijia1 aWang, Yixu1 aKuo, En-Jui1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2209.0457301519nas a2200133 4500008004100000245008200041210006900123260001500192520106200207100002601269700003001295700002301325856003701348 2023 eng d00aSubsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma0 aSubsystem CSS codes a tighter stabilizertoCSS mapping and Goursa c11/29/20233 a

The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly et. al. [quant-ph/0610153], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. We show that any subsystem stabilizer code can be ``doubled'' to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Leemhuis, and Terhal [New J. Phys. 12 083039 (2010)]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.

1 aLiu, Michael, Liaofan1 aTantivasadakarn, Nathanan1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2311.1800302547nas a2200205 4500008004100000245006500041210006400106260001400170490000900184520193000193653002702123653003102150100002002181700002202201700002302223700002202246700001702268700001902285856003702304 2023 eng d00aTime-energy uncertainty relation for noisy quantum metrology0 aTimeenergy uncertainty relation for noisy quantum metrology c12/5/20230 v4(4)3 a

Detection of weak forces and precise measurement of time are two of the many applications of quantum metrology to science and technology. We consider a quantum system initialized in a pure state and whose evolution is goverened by a Hamiltonian H; a measurement can later estimate the time t for which the system has evolved. In this work, we introduce and study a fundamental trade-off which relates the amount by which noise reduces the accuracy of a quantum clock to the amount of information about the energy of the clock that leaks to the environment. Specifically, we consider an idealized scenario in which Alice prepares an initial pure state of the clock, allows the clock to evolve for a time t that is not precisely known, and then transmits the clock through a noisy channel to Bob. The environment (Eve) receives any information that is lost. We prove that Bob's loss of quantum Fisher information (QFI) about t is equal to Eve's gain of QFI about a complementary energy parameter. We also prove a more general trade-off that applies when Bob and Eve wish to estimate the values of parameters associated with two non-commuting observables. We derive the necessary and sufficient conditions for the accuracy of the clock to be unaffected by the noise. These are a subset of the Knill-Laflamme error-correction conditions; states satisfying these conditions are said to form a metrological code. We provide a scheme to construct metrological codes in the stabilizer formalism. We show that there are metrological codes that cannot be written as a quantum error-correcting code with similar distance in which the Hamiltonian acts as a logical operator, potentially offering new schemes for constructing states that do not lose any sensitivity upon application of a noisy channel. We discuss applications of our results to sensing using a many-body state subject to erasure or amplitude-damping noise.

10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aFaist, Philippe1 aWoods, Mischa, P.1 aAlbert, Victor, V.1 aRenes, Joseph, M.1 aEisert, Jens1 aPreskill, John uhttps://arxiv.org/abs/2207.1370701108nas a2200145 4500008004100000245001300041210001300054260001500067520075200082100002200834700002100856700002500877700002300902856003700925 2023 eng d00aÆ codes0 aÆ codes c11/21/20233 a

Diatomic molecular codes [{arXiv:1911.00099}] are designed to encode quantum information in the orientation of a diatomic molecule, allowing error correction from small torques and changes in angular momentum. Here, we directly study noise native to atomic and molecular platforms -- spontaneous emission, stray electromagnetic fields, and Raman scattering -- and derive simple necessary and sufficient conditions for codes to protect against such noise. We identify existing and develop new absorption-emission (Æ) codes that are more practical than molecular codes, require lower average momentum, can directly protect against photonic processes up to arbitrary order, and are applicable to a broader set of atomic and molecular systems.

1 aJain, Shubham, P.1 aHudson, Eric, R.1 aCampbell, Wesley, C.1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2311.1232401196nas a2200133 4500008004100000245005300041210005100094260001500145490000800160520081600168100001800984700002301002856003701025 2022 eng d00aApproximating the two-mode two-photon Rabi model0 aApproximating the twomode twophoton Rabi model c01/17/20220 v4223 a

The Rabi model describes the simplest nontrivial interaction between a few-level system and a bosonic mode, featuring in multiple seemingly unrelated systems of importance to quantum science and technology. While exact expressions for the energies of this model and its few-mode extensions have been obtained, they involve roots of transcendental functions and are thus cumbersome and unintuitive. Utilizing the symmetric generalized rotating wave approximation (S-GRWA), we develop a family of approximations to the energies of the two-mode two-photon Rabi model. The simplest elements of the family are analytically tractable, providing good approximations in regimes of interest such as ultra- and deep-strong coupling. Higher-order approximate energies can be used if more accuracy is required. 

1 aWu, David, H.1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2012.0699400958nas a2200157 4500008004100000245004700041210004600088260001500134520045900149653004300608653002700651653003100678653003100709100002300740856003700763 2022 eng d00aBosonic coding: introduction and use cases0 aBosonic coding introduction and use cases c11/10/20223 a

Bosonic or continuous-variable coding is a field concerned with robust quantum information processing and communication with electromagnetic signals or mechanical modes. I review bosonic quantum memories, characterizing them as either bosonic stabilizer or bosonic Fock-state codes. I then enumerate various applications of bosonic encodings, four of which circumvent no-go theorems due to the intrinsic infinite-dimensionality of bosonic systems.

10aFOS: Computer and information sciences10aFOS: Physical sciences10aInformation Theory (cs.IT)10aQuantum Physics (quant-ph)1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2211.0571401146nas a2200169 4500008004100000245005900041210005900100260001400159300001100173490000800184520067200192100001900864700001500883700001800898700002300916856003700939 2022 eng d00aChiral central charge from a single bulk wave function0 aChiral central charge from a single bulk wave function c4/28/2022 a1764020 v1283 a

A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real-valued in any local product basis.

1 aKim, Isaac, H.1 aShi, Bowen1 aKato, Kohtaro1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2110.0693201776nas a2200181 4500008004100000245004200041210004100083260001400124520125200138653002701390653003101417100002301448700002301471700002001494700002201514700002101536856003701557 2022 eng d00aContinuous-Variable Shadow Tomography0 aContinuousVariable Shadow Tomography c11/9/20223 a

Shadow tomography is a framework for constructing succinct descriptions of quantum states, called classical shadows, with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable tomography in the classical-shadow framework, obtaining rigorous bounds on the sample complexity for estimating density matrices from these protocols. We analyze the efficiency of homodyne, heterodyne, photon number resolving (PNR), and photon-parity protocols. To reach a desired precision on the classical shadow of an N-photon density matrix with a high probability, we show that homodyne detection requires an order O(N5) measurements in the worst case, whereas PNR and photon-parity detection require O(N4) measurements in the worst case (both up to logarithmic corrections). We benchmark these results against numerical simulation as well as experimental data from optical homodyne experiments. We find that numerical and experimental homodyne tomography significantly outperforms our bounds, exhibiting a more typical scaling of the number of measurements that is close to linear in N. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.

10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aGandhari, Srilekha1 aAlbert, Victor, V.1 aGerrits, Thomas1 aTaylor, Jacob, M.1 aGullans, Michael uhttps://arxiv.org/abs/2211.0514901846nas a2200181 4500008004100000245009600041210006900137260001400206520121100220653003801431653004301469653002701512653003101539100001501570700001901585700002301604856003701627 2022 eng d00aGroup coset monogamy games and an application to device-independent continuous-variable QKD0 aGroup coset monogamy games and an application to deviceindepende c12/7/20223 a

We develop an extension of a recently introduced subspace coset state monogamy-of-entanglement game [Coladangelo, Liu, Liu, and Zhandry; Crypto'21] to general group coset states, which are uniform superpositions over elements of a subgroup to which has been applied a group-theoretic generalization of the quantum one-time pad. We give a general bound on the winning probability of a monogamy game constructed from subgroup coset states that applies to a wide range of finite and infinite groups. To study the infinite-group case, we use and further develop a measure-theoretic formalism that allows us to express continuous-variable measurements as operator-valued generalizations of probability measures.
We apply the monogamy game bound to various physically relevant groups, yielding realizations of the game in continuous-variable modes as well as in rotational states of a polyatomic molecule. We obtain explicit strong bounds in the case of specific group-space and subgroup combinations. As an application, we provide the first proof of one sided-device independent security of a squeezed-state continuous-variable quantum key distribution protocol against general coherent attacks.

10aCryptography and Security (cs.CR)10aFOS: Computer and information sciences10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aCulf, Eric1 aVidick, Thomas1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2212.0393501471nas a2200157 4500008004100000245005900041210005800100260001400158490000800172520102100180100001901201700001501220700001801235700002301253856003701276 2022 eng d00aModular commutator in gapped quantum many-body systems0 aModular commutator in gapped quantum manybody systems c8/26/20220 v1063 a

In arXiv:2110.06932, we argued that the chiral central charge -- a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system -- can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.7%.

1 aKim, Isaac, H.1 aShi, Bowen1 aKato, Kohtaro1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2110.1040002380nas a2200181 4500008004100000245007100041210006900112260001400181300000800195490000600203520186000209100001302069700002302082700002402105700001902129700001302148856003702161 2022 eng d00aProvably accurate simulation of gauge theories and bosonic systems0 aProvably accurate simulation of gauge theories and bosonic syste c9/20/2022 a8160 v63 a

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1).

1 aTong, Yu1 aAlbert, Victor, V.1 aMcClean, Jarrod, R.1 aPreskill, John1 aSu, Yuan uhttps://arxiv.org/abs/2110.0694201744nas a2200169 4500008004100000245007100041210006900112260001400181490000800195520123200203100002101435700001901456700002001475700002301495700001901518856003701537 2022 eng d00aProvably efficient machine learning for quantum many-body problems0 aProvably efficient machine learning for quantum manybody problem c9/26/20220 v3773 a

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

1 aHuang, Hsin-Yuan1 aKueng, Richard1 aTorlai, Giacomo1 aAlbert, Victor, V.1 aPreskill, John uhttps://arxiv.org/abs/2106.1262701481nas a2200193 4500008004100000245004300041210004200084260001400126300001100140490000800151520095100159100001501110700002001125700002301145700001801168700002001186700001701206856006401223 2021 eng d00aPhase-engineered bosonic quantum codes0 aPhaseengineered bosonic quantum codes c6/29/2021 a0624270 v1033 a

Continuous-variable systems protected by bosonic quantum codes have emerged as a promising platform for quantum information. To date, the design of code words has centered on optimizing the state occupation in the relevant basis to generate the distance needed for error correction. Here, we show tuning the phase degree of freedom in the design of code words can affect, and potentially enhance, the protection against Markovian errors that involve excitation exchange with the environment. As illustrations, we first consider phase engineering bosonic codes with uniform spacing in the Fock basis that correct excitation loss with a Kerr unitary and show that these modified codes feature destructive interference between error code words and, with an adapted “two-level” recovery, the error protection is significantly enhanced. We then study protection against energy decay with the presence of mode nonlinearities …

1 aLi, Linshu1 aYoung, Dylan, J1 aAlbert, Victor, V.1 aNoh, Kyungjoo1 aZou, Chang-Ling1 aJiang, Liang uhttps://authors.library.caltech.edu/109764/2/1901.05358.pdf01449nas a2200169 4500008004100000245007200041210006900113260001500182520093700197100002301134700001701157700001601174700001501190700001801205700001901223856003701242 2021 eng d00aSpin chains, defects, and quantum wires for the quantum-double edge0 aSpin chains defects and quantum wires for the quantumdouble edge c11/23/20213 a

Non-Abelian defects that bind Majorana or parafermion zero modes are prominent in several topological quantum computation schemes. Underpinning their established understanding is the quantum Ising spin chain, which can be recast as a fermionic model or viewed as a standalone effective theory for the surface-code edge -- both of which harbor non-Abelian defects. We generalize these notions by deriving an effective Ising-like spin chain describing the edge of quantum-double topological order. Relating Majorana and parafermion modes to anyonic strings, we introduce quantum-double generalizations of non-Abelian defects. We develop a way to embed finite-group valued qunits into those valued in continuous groups. Using this embedding, we provide a continuum description of the spin chain and recast its non-interacting part as a quantum wire via addition of a Wess-Zumino-Novikov-Witten term and non-Abelian bosonization.

1 aAlbert, Victor, V.1 aAasen, David1 aXu, Wenqing1 aJi, Wenjie1 aAlicea, Jason1 aPreskill, John uhttps://arxiv.org/abs/2111.1209602409nas a2200193 4500008004100000245006700041210006700108260001500175490000700190520183800197100002002035700001902055700002302074700001802097700002402115700002002139700001902159856003702178 2020 eng d00aContinuous symmetries and approximate quantum error correction0 aContinuous symmetries and approximate quantum error correction c10/26/20200 v103 a

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

1 aFaist, Philippe1 aNezami, Sepehr1 aAlbert, Victor, V.1 aSalton, Grant1 aPastawski, Fernando1 aHayden, Patrick1 aPreskill, John uhttps://arxiv.org/abs/1902.0771401214nas a2200145 4500008004100000245004500041210004500086260001300131490000700144520081700151100002300968700002100991700001901012856003701031 2020 eng d00aRobust Encoding of a Qubit in a Molecule0 aRobust Encoding of a Qubit in a Molecule c9/1/20200 v103 a

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of rotational states of a rigid body. These codes, which protect against both drift in the body’s orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules as well as nuclear states of molecules and atoms. We also describe codes associated with general non-Abelian groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

1 aAlbert, Victor, V.1 aCovey, Jacob, P.1 aPreskill, John uhttps://arxiv.org/abs/1911.0009901648nas a2200193 4500008004100000245006700041210006700108260001300175300001100188490000800199520108700207100001601294700001901310700002201329700001801351700002301369700002501392856003701417 2020 eng d00aSymmetry breaking and error correction in open quantum systems0 aSymmetry breaking and error correction in open quantum systems c8/6/2020 a2404050 v1253 a

Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system---a Lindbladian---a unitary symmetry can be imposed in a "weak" or a "strong" way. We characterize the possible Zn symmetry breaking transitions for both cases. In the case of Z2, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially-protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correctio

1 aLieu, Simon1 aBelyansky, Ron1 aYoung, Jeremy, T.1 aLundgren, Rex1 aAlbert, Victor, V.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2008.0281601534nas a2200169 4500008004100000245006200041210005800103260001500161490000800176520102300184100002301207700002201230700002301252700002501275700002701300856003701327 2017 eng d00aA solvable family of driven-dissipative many-body systems0 asolvable family of drivendissipative manybody systems c2017/11/100 v1193 a

Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, some of which can be viewed as dissipative analogues of the transverse-field Ising model, in that an effectively classical Hamiltonian is frustrated by dissipative processes that drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad and experimentally relevant subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in a many-body model that can be realized naturally with Rydberg atoms or trapped ions, and to compute the effects of decoherence on a canonical trapped-ion-based quantum computation architecture.

1 aFoss-Feig, Michael1 aYoung, Jeremy, T.1 aAlbert, Victor, V.1 aGorshkov, Alexey, V.1 aMaghrebi, Mohammad, F. uhttps://arxiv.org/abs/1703.04626