A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.
The possibility of stochastic resonance of a quantum channel and hence the noise enhanced quantum channel capacity is explored by considering one-Pauli channels which are more classical like. The fidelity of the channel is also considered.
The sufficient condition of entanglement enhanced classical capacity is given for Pauli memory channel with arbitrary channel parameters. In some special case the condition is also necessary but fail to be necessary in general. The theory of majorization and perturbation are used in the proving.
The optimal measurement configuration, i.e., the optimal input quantum state and measurement in the form of a POVM with two elements, is investigated in this paper for qubit and generalized Pauli channels. The channel directions are defined as the contracting directions of the channel with an arbitrary fixed basis of Pauli- matrices. In the qubit Pauli channel case with known channel directions it is shown that the optimal configuration that maximizes the Fisher information of the estimated parameters consists of three von Neumann measurements together with pure input states directed to the ap…
A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fishe…
We propose a parameter estimation protocol for generalized Pauli channels acting on $d$-dimensional Hilbert space. The salient features of the proposed method include product probe states and measurements, the number of measurement configurations linear in $d$, minimal post-processing, and the scaling of the mean square error comparable to that of the entanglement-based parameter estimation scheme for generalized Pauli channels. We also show that while measuring generalized Pauli operators the errors caused by the Pauli noise can be modeled as measurement errors. This makes it possible to util…
We analyze the quantum evolution represented by a time-dependent family of generalized Pauli channels. This evolution is provided by the random decoherence channels with respect to the maximal number of mutually unbiased bases. We derive the necessary and sufficient conditions for the vanishing back-flow of information.
We introduce graph state basis diagonalization to calculate the coherent information of a quantum code passing through a Pauli channel. The scheme is 5000 times faster than the best known one for some concatenated repetition codes, providing us a practical constructive way of approaching the quantum capacity of a Pauli channel. The calculation of the coherent information of non-additive quantum code can also be greatly simplified in graph state basis.
We consider the problem of optimally discriminating two Pauli channels in the minimax strategy, maximizing the smallest of the probabilities of correct identification of the channel. We find the optimal input state at the channel and show the conditions under which using entanglement strictly enhances distinguishability. We finally compare the minimax strategy with the Bayesian one.
We prove that a general upper bound on the maximal mutual information of quantum channels is saturated in the case of Pauli channels with an arbitrary degree of memory. For a subset of such channels we explicitly identify the optimal signal states. We show analytically that for such a class of channels entangled states are indeed optimal above a given memory threshold. It is noteworthy that the resulting channel capacity is a non-differentiable function of the memory parameter.
We consider the simulation of a quantum channel by two parties who share a resource state and may apply local operations assisted by classical communication (LOCC). One specific type of such LOCC is standard teleportation, which is however limited to the simulation of Pauli channels. Here we show how we can easily enlarge this class by means of a minimal perturbation of the teleportation protocol, where we introduce noise in the classical communication channel between the remote parties. By adopting this noisy protocol, we provide a necessary condition for simulating a non-Pauli channel. In pa…
Pauli blocking of spontaneous emission is responsible for the stability of atoms. Higher electronic orbitals cannot decay to lower-lying states if they are already occupied -- this is Pauli blocking due to occupation of internal states. Pauli blocking also occurs when free atoms scatter light elastically (Rayleigh scattering) and the final external momentum states are already occupied. A suppression of the total rate of light scattering requires a quantum-degenerate Fermi gas with a Fermi energy larger than the photon recoil energy. This has been predicted more than 30 years ago, but never rea…
Finite-time Markovian channels, unlike their infinitesimal counterparts, do not form a convex set. As a particular instance of this observation, we consider the problem of mixing the three Pauli channels, conservatively assumed to be quantum dynamical semigroups, and fully characterize the resulting ``Pauli simplex.'' We show that neither the set of non-Markovian (completely positive indivisible) nor Markovian channels is convex in the Pauli simplex, and that the measure of non-Markovian channels is about 0.87. All channels in the Pauli simplex are P divisible. A potential application in the c…
The Gottesman-Knill theorem allows for the efficient simulation of stabilizer-based quantum error-correction circuits. Errors in these circuits are commonly modeled as depolarizing channels by using Monte Carlo methods to insert Pauli gates randomly throughout the circuit. Although convenient, these channels are poor approximations of common, realistic channels like amplitude damping. Here we analyze a larger set of efficiently simulable error channels by allowing the random insertion of any one-qubit gate or measurement that can be efficiently simulated within the stabilizer formalism. Our ne…
Twirling is a technique widely used for converting arbitrary noise channels into Pauli channels in error threshold estimations of quantum error correction codes. It is vitally useful both in real experiments and in classical quantum simulations. Minimising the size of the twirling gate set increases the efficiency of simulations and in experiments it might reduce both the number of runs required and the circuit depth (and hence the error burden). Conventional twirling uses the full set of Pauli gates as the set of twirling gates. This article provides a theoretical background for Pauli twirlin…
We analyze the fidelity of the generalized Pauli channels governed by memory kernel master equations. It is shown that, by appropriate engineering of parameters of the corresponding memory kernel, the quantum evolution with non-local noise can have higher fidelity than the corresponding purely Markovian evolution governed by the Markovian semigroup. Similar engineering can substantially influence the evolution of quantum entanglement, entropy, and quantum coherence.
A family of quantum cloning machines is introduced that produce two approximate copies from a single quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, describing the balance between the quality of the two copies. This also provides an upper bound on the quantum capacity of the Pauli channel with probabilities $p_x$, $p_y$ and $p_z$. The capacity is shown to be vanishing if $(\sqrt{p_x},\sqrt{p_y},\sqrt{p_z})$ lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal clon…
We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the opt…
We study the memory property of the channels obtained by convex combinations of Markovian channels that are not necessarily quantum dynamical semigroups (QDSs). In particular, we characterize the geometry of the region of (non-)Markovian channels obtained by the convex combination of the three Pauli channels, as a function of deviation from the semigroup form in a family of channels. The regions are highly convex, and interestingly, the measure of the non-Markovian region shrinks with greater deviation from the QDS structure for the considered family, underscoring the counterintuitive nature o…
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here we prove several results on efficiently learning Pauli channels, and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on $n$ qubits with high probability to a relative precision $ε$ using $O\bigl(ε^{-2} n 2^n\bigr)$ measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measu…
Decoherence of quantum systems is described by quantum channels. However, a complete understanding of such channels, especially in the multi-particle setting, is still an ongoing difficult task. We propose the family of quantum maps that preserve or completely erase the components of a multi-qubit system in the basis of Pauli strings, which we call Pauli component erasing maps. For the corresponding channels, it is shown that the preserved components can be interpreted as a finite vector subspace, from which we derive several properties and complete the characterization. Moreover, we show that…
A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary trade-offs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations, and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some …
We present the experimental realization of the optimal estimation protocol for a Pauli noisy channel. The method is based on the generation of 2-qubit Bell states and the introduction of quantum noise in a controlled way on one of the state subsystems. The efficiency of the optimal estimation, achieved by a Bell measurement, is shown to outperform quantum process tomography.
We study encodings that give the best known thresholds for the non-zero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a non-zero capacity for a fidelity (probability of no error) of f=0.80870.
Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of c…
Single snapshot non-Markovianity of Pauli channels
IBM Quantum, IBM T. J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；CMC Microsystems（CMC微系统公司）；IBM Quantum, IBM Almaden Research Center（IBM量子，IBM阿尔马登研究中心）
We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, $Φ= e^{{\cal L}t}$,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together wit…
In this paper, we analyze the evolution of the generalized Pauli channels governed by the memory kernel master equation. We provide necessary and sufficient conditions for the memory kernel to give rise to the legitimate (completely positive and trace-preserving) quantum evolution. In particular, we analyze a class of kernels generating the quantum semi-Markov evolution, which is a natural generalization of the Markovian semigroup. Interestingly, the convex combination of Markovian semigroups goes beyond the semi-Markov case. Our analysis is illustrated with several examples.
The calculating of the coherent information is a fundamental step in obtaining the quantum capacity of a quantum channel. We introduce orthogonal and complete code basis to evaluate the coherent information per channel use when the input is the maximal mixture of stabilizer codewords. In the code basis, the output density matrix is diagonal, the joint output of the system and the auxiliary is block diagonal. The coherent information is worked out by counting the weights of error operators.
Quantum non-Markovianity of channels can be produced by mixing Markovian channels, as observed recently by various authors. We consider an analogous question of whether singularities of the channel can be produced by mixing non-singular channels, i.e., ones that lack them. Here we answer the question in the negative in the context of qubit Pauli channels. On the other hand, mixing channels with a singularity can lead to the elimination of singularities in the resultant channel. We distinguish between two types of singular channels, which lead under mixing to broadly quite different properties …
We study a class of qubit non-Markovian general Pauli dynamical maps with multiple singularities in the generator. We discuss a few easy examples involving trigonometric or other non-monotonic time dependence of the map, and discuss in detail the structure of channels which don't have any trigonometric functional dependence. We demystify the concept of a singularity here, showing that it corresponds to a point where the dynamics can be regular but the map is momentarily non-invertible, and this gives a basic guideline to construct such non-invertible non-Markovian channels. Most members of the…
We analyze the geometrical properties of trace-preserving Pauli maps. Using the Choi-Jamio\lkowski isomorphism, we express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the Pauli map. We analytically compute the volumes of positive, trace-preserving Pauli maps and Pauli channels. In particular, we find the relative volumes of the entanglement breaking Pauli channels, as well as the channels that can be generated by a time-local generator. Finally, we show what part of the Pauli channels are P and CP-divisible, which is related to the notion of Markovianity.
We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an $n$-qubit Pauli channel to $\pm\varepsilon$ precision. We give an estimation protocol with an $n$-qubit ancilla that succeeds with high probability using only $O(n/\varepsilon^{2})$ copies of the Pauli channel, while prove that any ancilla-free protocol (possibly with adaptive …
One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X , Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number eg of generic Pauli errors, plus eZ Pauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of err…
The accuracy of any physical scheme used to estimate the parameter describing the strength of a single qubit Pauli channel can be quantified using standard techniques from quantum estimation theory. It is known that the optimal estimation scheme, with m channel invocations, uses initial states for the systems which are pure and unentangled and provides an uncertainty of O[1/m^(1/2)]. This protocol is analogous to a classical repetition and averaging scheme. We consider estimation schemes where the initial states available are not pure and compare a protocol involving quantum correlated states …
We define and study the properties of channels which are analogous to unital qubit channels in several ways. A full treatment can be given only when the dimension d is a prime power, in which case each of the (d+1) mutually unbiased bases (MUB) defines an axis. Along each axis the channel looks like a depolarizing channel, but the degree of depolarization depends on the axis. When d is not a prime power, some of our results still hold, particularly in the case of channels with one symmetry axis. We describe the convex structure of this class of channels and the subclass of entanglement breakin…
Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an $n$-qubit channel to precision $ε$ in $\ell_\infty$ using just $O(1/ε^2) \log(n/ε)$ applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an $O(…
We examine the efficiency of pure, nondegenerate quantum-error correction-codes for Pauli channels. Specifically, we investigate if correction of multiple errors in a block is more efficient than using a code that only corrects one error per block. Block coding with multiple-error correction cannot increase the efficiency when the qubit error-probability is below a certain value and the code size fixed. More surprisingly, existing multiple-error correction codes with a code length equal or less than 256 qubits have lower efficiency than the optimal single-error correcting codes for any value o…
We analyze the geometry of the generalized Pauli channels constructed from the mutually unbiased bases. The Choi-Jamiołkowski isomorphism allows us to express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the generalized Pauli maps. After determining appropriate regions of integration, we analytically compute the volume of generalized Pauli channels and their important subclasses. In particular, we obtain the volumes of the generalized Pauli channels that can be generated by a legitimate generator and are entanglement breaking. We also provide the upper bound for …
In this paper we estimate the parameters of the qubit Pauli channel using the channel matrix formalism. The main novelty of this work is that we do not assume the directions of the Pauli channel to be known, but they are determined through the tomography process, too. The results show that for optimally estimating the contraction parameters and the channel matrix we should have input qubits and measurements in the channel directions. However, for optimally estimating the channel directions, we should use different tomography conditions.
We analyze the geometry of positive and completely positive, trace preserving Pauli maps that are fully determined by up to two distinct parameters. This includes five classes of symmetric and non-invertible Pauli channels. Using the Hilbert-Schmidt metric in the space of the Choi-Jamio\lkowski states, we compute the relative volumes of entanglement breaking, time-local generated, and divisible channels. Finally, we find the shapes of the complete positivity regions in relation to the tetrahedron of all Pauli channels.
We study the dynamics of four-qubit W state under various noisy environments by solving analytically the master equation in the Lindblad form in which the Lindblad operators correspond to the Pauli matrices and describe the decoherence of states. Also, we investigate the dynamics of the entanglement using the lower bound to the concurrence. It is found that while the entanglement decreases monotonically for Pauli-Z noise, it decays suddenly for other three noises. Moreover, by studying the time evolution of entanglement of various maximally entangled four-qubit states, we indicate that the fou…
We study entanglement dynamics of the three-qubit system which is initially prepared in pure Greenberger-Horne- Zeilinger (GHZ) or W state and transmitted through one of the Pauli channels $σ_z, \, σ_x, \, σ_y$ or the depolarizing channel. With the help of the lower bound for three-qubit concurrence we show that the W state preserves more entanglement than the GHZ state in transmission through the Pauli channel $σ_z$. For the Pauli channels $σ_x, \, σ_y$ and the depolarizing channel, however, the entanglement of the GHZ state is more resistant against decoherence than the W-type entanglement. …
We consider the asymptotic regularization of the maximal fidelity for the generalized Pauli channels, which is a problem similar to the classical channel capacity. In particular, we find the formulas for the extremal channel fidelities and the maximal output $\infty$-norm. For wide classes of channels, we show that these quantities are multiplicative. Finally, we find the regularized maximal fidelity for the channels satisfying the time-local master eqeuations.
