arXiv subjects

Victor V. Albert

Publications and source records attributed to Victor V. Albert.

At least 55 records · Page 3Linked to original sources

arXiv2023

Precision Bounds on Continuous-Variable State Tomography using Classical Shadows

Shadow tomography is a framework for constructing succinct descriptions of quantum states using randomized measurement bases, called classical shadows, with powerful methods to bound the estimators used. We recast existing experimental protocols for continuous-variable quantum state tomography in the classical-shadow framework, obtaining rigorous bounds on the number of independent measurements needed 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 $\mathcal{O}(N^{4+1/3})$ measurements in the worst case, whereas PNR and photon-parity detection require $\mathcal{O}(N^4)$ 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.

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arXiv2023

Quantum spherical codes

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.

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arXiv2023

Qubit-oscillator concatenated codes: decoding formalism & code comparison

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 there are several bosonic codes and concatenation schemes to choose from, including the recently discovered GKP-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)}] that allow protection of a logical bosonic mode from fluctuations of the mode's conjugate variables. 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/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.

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arXiv2022

Group coset monogamy games and an application to device-independent continuous-variable QKD

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.

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arXiv2022

Bosonic coding: introduction and use cases

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.

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arXiv2022

Provably efficient machine learning for quantum many-body problems

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.

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arXiv2022

Provably accurate simulation of gauge theories and bosonic systems

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 $Λ=\textrm{polylog}(ε^{-1})$.

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arXiv2021

Spin chains, defects, and quantum wires for the quantum-double edge

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.

cond-mat.str-el↗

arXiv2021

Approximating the two-mode two-photon Rabi model

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.

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arXiv2021

Modular commutator in gapped quantum many-body systems

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%.

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arXiv2021

Chiral central charge from a single bulk wave function

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.

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arXiv2020

Symmetry breaking and error correction in open quantum systems

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 $\mathbb{Z}_n$ symmetry breaking transitions for both cases. In the case of $\mathbb{Z}_2$, 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 correction.

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arXiv2019

Robust encoding of a qubit in a molecule

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert state 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 nonabelian compact Lie groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

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arXiv2019

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

This work derives an analytical formula for the asymptotic state---the quantum state resulting from an infinite number of applications of a general quantum channel on some initial state. For channels admitting multiple fixed or rotating points, conserved quantities---the left fixed/rotating points of the channel---determine the dependence of the asymptotic state on the initial state. The formula stems from a Noether-like theorem stating that, for any channel admitting a full-rank fixed point, conserved quantities commute with that channel's Kraus operators up to a phase. The formula is applied to adiabatic transport of the fixed-point space of channels, revealing cases where the dissipative/spectral gap can close during any segment of the adiabatic path. The formula is also applied to calculate expectation values of noninjective matrix product states (MPS) in the thermodynamic limit, revealing that those expectation values can also be calculated using an MPS with reduced bond dimension and a modified boundary.

quant-ph↗

arXiv2019

Continuous symmetries and approximate quantum error correction

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.

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arXiv2019

Optimized Entanglement Purification

We investigate novel protocols for entanglement purification of qubit Bell pairs. Employing genetic algorithms for the design of the purification circuit, we obtain shorter circuits achieving higher success rates and better final fidelities than what is currently available in the literature. We provide a software tool for analytical and numerical study of the generated purification circuits, under customizable error models. These new purification protocols pave the way to practical implementations of modular quantum computers and quantum repeaters. Our approach is particularly attentive to the effects of finite resources and imperfect local operations - phenomena neglected in the usual asymptotic approach to the problem. The choice of the building blocks permitted in the construction of the circuits is based on a thorough enumeration of the local Clifford operations that act as permutations on the basis of Bell states.

quant-ph↗

arXiv2019

Designing good bosonic quantum codes via creating destructive interference

Continuous-variable systems protected by bosonic quantum error-correcting codes have emerged as a promising platform for quantum information processing. To date, design of codewords has centered on optimizing the occupation of basis states in the error-relevant basis. Here, we propose utilizing the phase degree of freedom in basis state probability amplitudes to devise codes that feature destructive interference, and thus reduced overlap, between error codewords. To showcase, we first consider the correction of excitation loss using single-mode codes with Fock-space parity structure and show that, with a tailored "two-level" recovery, altering the signs of probability amplitudes can significantly suppress decoherence. We then study the joint channel of excitation loss and Kerr effect, and show the critical role of nontrivial phase for optimal quantum codes for such intricate yet important channels. The principle is extended to improve bosonic codes defined in other bases and multi-qubit codes, showing its wide applicability in quantum error correction.

quant-ph↗

arXiv2018

Pair-cat codes: autonomous error-correction with low-order nonlinearity

We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a two-dimensional complex plane, instead of the full four-dimensional two-mode phase space. The two-mode codes are generalized to multiple modes in an extension of the stabilizer formalism to non-diagonalizable stabilizers. The $M$-mode codes can protect against either arbitrary photon losses in up to $M-1$ modes or arbitrary losses and gains in any one mode.

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