Approximate Quantum Error Correction Research Articles

Quantifying noncovariance of quantum channels with respect to groups

A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.

Approximate Autonomous Quantum Error Correction with Reinforcement Learning.

Autonomous quantum error correction (AQEC) protects logical qubits by engineered dissipation and thus circumvents the necessity of frequent, error-prone measurement-feedback loops. Bosonic code spaces, where single-photon loss represents the dominant source of error, are promising candidates for AQEC due to their flexibility and controllability. While existing proposals have demonstrated the in-principle feasibility of AQEC with bosonic code spaces, these schemes are typically based on the exact implementation of the Knill-Laflamme conditions and thus require the realization of Hamiltonian distances d≥2. Implementing such Hamiltonian distances requires multiple nonlinear interactions and control fields, rendering these schemes experimentally challenging. Here, we propose a bosonic code for approximate AQEC by relaxing the Knill-Laflamme conditions. Using reinforcement learning (RL), we identify the optimal bosonic set of code words (denoted here by RL code), which, surprisingly, is composed of the Fock states |2⟩ and |4⟩. As we show, the RL code, despite its approximate nature, successfully suppresses single-photon loss, reducing it to an effective dephasing process that well surpasses the break-even threshold. It may thus provide a valuable building block toward full error protection. The error-correcting Hamiltonian, which includes ancilla systems that emulate the engineered dissipation, is entirely based on the Hamiltonian distance d=1, significantly reducing model complexity. Single-qubit gates are implemented in the RL code with a maximum distance d_{g}=2.

Approximate quantum error correction, covariance symmetry, and their relation

Approximate quantum error correction, covariance symmetry, and their relation

Quantifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT-extendible channels

The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semi-definite relaxation of this optimization task by instead optimizing over the larger set of two-PPT-extendible channels. The main analytical calculations in our paper consist of exploiting the unitary covariance symmetry of the identity channel to establish a significant reduction of the computational cost of this latter optimization. Next, by exploiting known connections between approximate teleportation and quantum error correction, we also apply these concepts to establish bounds on the performance of approximate quantum error correction over a given quantum channel. Finally, we evaluate our bounds for various examples of resource states and channels.

Linear Programming Bounds for Approximate Quantum Error Correction Over Arbitrary Quantum Channels

While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary quantum channels that describe bespoke noise models. Herein, for any Kraus decomposition of any given quantum channel, we introduce corresponding quantum weight enumerators that naturally generalize the Shor-Laflamme quantum weight enumerators. We establish an indirect linear relationship between these generalized quantum weight enumerators by introducing an auxiliary exact weight enumerator that completely quantifies the quantum code's projector, and is independent of the underlying noise process. By additionally working within the framework of approximate quantum error correction, we establish a general framework for constructing a linear program that is infeasible whenever approximate quantum error correcting codes with corresponding parameters do not exist. Our linear programming framework allows us to establish the non-existence of certain quantum codes that approximately correct amplitude damping errors, and obtain non-trivial upper bounds on the maximum dimension of a broad family of permutation-invariant quantum codes.

Quantum-enhanced radiometry via approximate quantum error correction

Quantum sensing based on exotic quantum states is appealing for practical metrology applications and fundamental studies. However, these quantum states are vulnerable to noise and the resulting quantum enhancement is weakened in practice. Here, we experimentally demonstrate a quantum-enhanced sensing scheme with a bosonic probe, by exploring the large Hilbert space of the bosonic mode and developing both the approximate quantum error correction and the quantum jump tracking approaches. In a practical radiometry scenario, we attain a 5.3 dB enhancement of sensitivity, which reaches 9.1 × 10−4 Hz−1/2 when measuring the excitation population of a receiver mode. Our results demonstrate the potential of quantum sensing with near-term quantum technologies, not only shedding new light on the quantum advantage of sensing, but also stimulating further efforts on bosonic quantum technologies.

Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries

Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of fault-tolerant quantum computing), in contrast to the case without symmetry constraints. Furthermore, several quantitative fundamental limits on the accuracy of such covariant codes for approximate quantum error correction are known. Here, we consider the quantum error correction capability of uniformly random covariant codes. In particular, we analytically study the most essential cases of $U(1)$ and $SU(d)$ symmetries, and show that for both symmetry groups the error of the covariant codes generated by Haar-random symmetric unitaries, i.e., unitaries that commute with the group actions, typically scale as $O(n^{-1})$ in terms of both the average- and worst-case purified distances against erasure noise, saturating the fundamental limits to leading order. We note that the results hold for symmetric variants of unitary 2-designs, and comment on the convergence problem of symmetric random circuits. Our results not only indicate (potentially efficient) randomized constructions of optimal $U(1)$- and $SU(d)$-covariant codes, but also reveal fundamental properties of random symmetric unitaries, which yield important solvable models of complex quantum systems (including black holes and many-body spin systems) that have attracted great recent interest in quantum gravity and condensed matter physics. We expect our construction and analysis to find broad relevance in both physics and quantum computing.

Approximate Petz Recovery from the Geometry of Density Operators

We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched R\'{e}nyi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$\epsilon$ accuracy whenever the data processing inequality for $\tilde{D}_{2}$ is saturated up to terms of order $\epsilon^2$ times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched R\'{e}nyi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched R\'{e}nyi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.

Hyperinvariant multiscale entanglement renormalization ansatz: Approximate holographic error correction codes with power-law correlations

We consider a class of holographic tensor networks that are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility for an efficient variational ansatz, we provide guidelines for constructing networks consisting of multiple types of tensors that can support power-law correlation. We also provide an explicit construction of one such network, which approximates the holographic HaPPY pentagon code in the limit where variational parameters are taken to be small.

From Quantum Codes to Gravity: A Journey of Gravitizing Quantum Mechanics

In this note, I review a recent approach to quantum gravity that “gravitizes” quantum mechanics by emerging geometry and gravity from complex quantum states. Drawing further insights from tensor network toy models in AdS/CFT, I propose that approximate quantum error correction codes, when re-adapted into the aforementioned framework, also have promise in emerging gravity in near-flat geometries.

Trade-Offs on Number and Phase Shift Resilience in Bosonic Quantum Codes

Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code utilizes is helpful. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer g apart, and the gap g imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting g-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, g-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing and amplitude damping errors.

Semidefinite programming hierarchies for constrained bilinear optimization

We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form {mathrm {Tr}}big [H(Dotimes E)big ], maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(d^{k/2}) to {mathrm {poly}}(d,k), where d denotes local dimension and k the number of copies.

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.

Optimal approximate quantum error correction for quantum metrology

The authors study the estimation of a Hamiltonian parameter under Markovian noise with the help of fast quantum controls. When the sensing time is long, the paper shows that for a generic set of Markovian evolutions, there exists an approximate quantum error correction strategy which is the best among all types of quantum controls and minimizes the estimation error asymptotically.

Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit

We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them, replacing classical bits by zero-bits makes teleportation asymptotically reversible. The decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applications to state merging, the quantum channel capacity, entanglement distillation, quantum identification and remote state preparation. The source of these results is the theory of approximate quantum error correction. The action of a quantum channel is reversible if and only if no information is leaked to the environment, a characterization that is useful even in approximate form. However, different notions of approximation lead to qualitatively different forms of quantum error correction in the limit of large dimension. We study the effect of a constraint on the dimension of the reference system when considering information leakage. While the resulting condition fails to ensure that the entire input can be corrected, it does ensure that all subspaces of dimension matching that of the reference are correctable. The size of the reference can be characterized by a parameter $\alpha$, we call the associated resource an $\alpha$-bit. Changing $\alpha$ interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of $\alpha$-bits, including the applications above, and determine the $\alpha$-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants.

Eigenstate thermalization hypothesis and approximate quantum error correction

The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.

Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy

The data processing inequality states that the quantum relative entropy between two states rho and sigma can never increase by applying the same quantum channel mathcal {N} to both states. This inequality can be strengthened with a remainder term in the form of a distance between rho and the closest recovered state (mathcal {R} circ mathcal {N})(rho ), where mathcal {R} is a recovery map with the property that sigma = (mathcal {R} circ mathcal {N})(sigma ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on sigma and the quantum channel mathcal {N} to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

Towards Holography via Quantum Source-Channel Codes.

While originally motivated by quantum computation, quantum error correction (QEC) is currently providing valuable insights into many-body quantum physics, such as topological phases of matter. Furthermore, mounting evidence originating from holography research (AdS/CFT) indicates that QEC should also be pertinent for conformal field theories. With this motivation in mind, we introduce quantum source-channel codes, which combine features of lossy compression and approximate quantum error correction, both of which are predicted in holography. Through a recent construction for approximate recovery maps, we derive guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information. As an example, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they exhibit nontrivial protection from local erasure. This gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code. We argue that quantum source-channel codes are of independent interest beyond holography.

Uncertainty relations and approximate quantum error correction

The uncertainty principle can be understood as constraining the probability of winning a game in which Alice measures one of two conjugate observables, such as position or momentum, on a system provided by Bob, and he is to guess the outcome. Two variants are possible: either Alice tells Bob which observable she measured, or he has to furnish guesses for both cases. Here I derive new uncertainty relations for both, formulated directly in terms of Bob's guessing probabilities. For the former these relate to the entanglement that can be recovered by action on Bob's system alone. This gives a condition for approximate quantum error correction in terms of the recoverability of "amplitude" and "phase" information, implicitly used in the recent construction of efficient quantum polar codes. I also find a tight relation on the guessing probabilities for the latter game, which has application to wave-particle duality relations.

New Class of Quantum Error-Correcting Codes for a Bosonic Mode

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.