Discrete Wigner Function Research Articles

Critical behaviors of non-stabilizerness in quantum spin chains

Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.

Protecting quantum correlations of negative quantum states using weak measurement under non-Markovian noise

The weak measurement (WM) and quantum measurement reversal (QMR) are crucial in protecting the collapse of quantum states. The idea of WM and QMR has recently been used to protect and enhance quantum correlations and universal quantum teleportation (UQT) protocols. Here, we study the quantum correlations, maximal fidelity, and fidelity deviation of the two-qubit negative quantum states developed using discrete Wigner functions with(without) WM and QMR. To take into account the effect of a noisy environment, we evolve the states via non-Markovian amplitude damping and random telegraph noise quantum channels. To benchmark the performance of negative quantum states, we calculate their success probability. We compare our results with the two-qubit maximally entangled Bell state. Interestingly, we observe that some negative quantum states perform better with WM and QMR than the Bell state for different cases under evolution via noisy quantum channels.

Connecting Continuous and Discrete Wigner Functions Via GKP Encoding

Connecting Continuous and Discrete Wigner Functions Via GKP Encoding

Tomographic Universality of the Discrete Wigner Function

We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup.

Harnessing Quantumness of States using Discrete Wigner Functions under (non)‐Markovian Quantum Channels

AbstractThe negativity of the discrete Wigner functions (DWFs) is a measure of non‐classicality and is often used to quantify the degree of quantum coherence in a system. The study of Wigner negativity and its evolution under different quantum channels can provide insight into the stability and robustness of quantum states under their interaction with the environment, which is essential for developing practical quantum computing systems. The variation of DWF negativity of qubit, qutrit, and two‐qubit systems under the action of (non)‐Markovian random telegraph noise (RTN) and amplitude damping (AD) quantum channels is investigated. Different negative quantum states that can be used as a resource for quantum computation and quantum teleportation are constructed. The success of quantum computation and teleportation is estimated for these states under (non)‐Markovian evolutions.

Detecting quantum phase transition via magic resource in the XY spin model

Quantum phase transition in the $XY$ spin model with three-spin interaction is investigated using magic resource (non-stabilizerness), which is crucial in universal fault-tolerant quantum computation. The magic quantifier we employ here is defined straightforwardly via characteristic functions of quantum states, which are well defined for all dimensional quantum systems (in sharp contrast to those defined by discrete Wigner functions) and can be easily calculated. We show that the magic quantifier of both the reduced single-site spins and two-site spins of the system ground state increase to their maximum around the critical points for quantum phase transition. This indicates that the magic resource can be used to detect the critical phenomena in the $XY$ spin model and reveals a connection between quantum phase transition in many-body systems and quantum resource in stabilizer quantum computation.

Uniqueness of Noncontextual Models for Stabilizer Subtheories.

We give a complete characterization of the (non)classicality of all stabilizer subtheories. First, we prove that there is a unique nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory in all odd dimensions, namely Gross's discrete Wigner function. This representation is equivalent to Spekkens' epistemically restricted toy theory, which is consequently singled out as the unique noncontextual ontological model for the stabilizer subtheory. Strikingly, the principle of noncontextuality is powerful enough (at least in this setting) to single out one particular classical realist interpretation. Our result explains the practical utility of Gross's representation by showing that (in the setting of the stabilizer subtheory) negativity in this particular representation implies generalized contextuality. Since negativity of this particular representation is a necessary resource for universal quantum computation in the state injection model, it follows that generalized contextuality is also a necessary resource for universal quantum computation in this model. In all even dimensions, we prove that there does not exist any nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory, and, hence, that the stabilizer subtheory is contextual in all even dimensions.

Quantum Advantage in Information Retrieval

Random access codes have provided many examples of quantum advantage in communication, but concern only one kind of information retrieval task. We introduce a related task - the Torpedo Game - and show that it admits greater quantum advantage than the comparable random access code. Perfect quantum strategies involving prepare-and-measure protocols with experimentally accessible three-level systems emerge via analysis in terms of the discrete Wigner function. The example is leveraged to an operational advantage in a pacifist version of the strategy game Battleship. We pinpoint a characteristic of quantum systems that enables quantum advantage in any bounded-memory information retrieval task. While preparation contextuality has previously been linked to advantages in random access coding we focus here on a different characteristic called sequential contextuality. It is shown not only to be necessary and sufficient for quantum advantage, but also to quantify the degree of advantage. Our perfect qutrit strategy for the Torpedo Game entails the strongest type of inconsistency with non-contextual hidden variables, revealing logical paradoxes with respect to those assumptions.

Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we employ a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths. This model is an ideal platform not only for a periodic system but also for a non-periodic system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. As demonstrations, we numerically integrate the discrete QFPE for a 2D free rotor and harmonic potential systems in a high-temperature Markovian case using a coarse mesh with initial conditions that involve singularity.

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the π8 gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on t qutrit π8 gates that are followed by Clifford evolution, and show that this only requires 3t2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.

Transition probabilities and transition rates in discrete phase space

The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.

Discrete Wigner functions from informationally complete quantum measurements

Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing a new way in which the symmetric informationally complete measurements (SICs) are significant. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements conceptually underlying the associated quasiprobability representations.

On the discrete Wigner function for

We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.

Discrete and generalized phase space techniques in critical quantum spin chains

We apply the Wigner function formalism from quantum optics via two approaches, Wootters' discrete Wigner function and the generalized Wigner function, to detect quantum phase transitions in critical spin-12 systems. We develop a general formula relating the phase space techniques and the thermodynamical quantities of spin models, which we apply to single, bipartite and multi-partite systems governed by the XY and the XXZ models. Our approach allows us to introduce a novel way to represent, detect, and distinguish first-, second- and infinite-order quantum phase transitions. Furthermore, we show that the factorization phenomenon of the XY model is only directly detectable by quantities based on the square root of the bipartite reduced density matrix. We establish that phase space techniques provide a simple, experimentally promising tool in the study of many-body systems and we discuss their relation with measures of quantum correlations and quantum coherence.

Simulation of qubit quantum circuits via Pauli propagation

We present novel algorithms to estimate outcomes for qubit quantum circuits. Notably, these methods can simulate a Clifford circuit in linear time without ever writing down stabilizer states explicitly. These algorithms outperform previous noisy near-Clifford techniques for most circuits. We identify a large class of input states that can be efficiently simulated despite not being stabilizer states. The algorithms leverage probability distributions constructed from Bloch vectors, paralleling previously known algorithms that use the discrete Wigner function for qutrits.

Dynamical symmetry in a minimal dimeric complex

The emergence of non-configurational symmetry is studied in a minimal example. The system under scrutiny consists of a dimeric hexagonal complex with configurational C3 symmetry, formulated as a tight-binding model. An accidental three-fold degeneracy point in parameter space is found; it is shown that an internal symmetry group operates on Hilbert space, but not on configuration space. The corresponding discrete Wigner functions for the irreducible representations of are utilized to show that a phase space is sufficient to exhibit an invariant subset. The dynamical symmetry is thus identified with a discrete semi-plane. Some implications on other known hidden symmetries of continuous systems are qualitatively discussed.

The Negativity‐to‐Violation Map between Wigner Function and Quantum Contextuality Inequality for a Single Qudit

AbstractThe relation between discrete Wigner function and quantum contextuality based on graph theory has been studied, following the work in [Nature 510,351(2014)]. To do this, non‐stabilizer projectors have been introduced to a series of non‐contextuality graphs based on stabilizer projectors for a single qudit with odd prime dimension. It has been found that, for a phase space point defined by Wootters, there exists a given set of states for an odd‐prime qudit where the negative discrete Wigner function on the phase space point means its quantum contextuality under measurements on the graphs designed by a specific method. To implement this method, a subset of non‐stabilizer projectors has been found. In the union of the set of states for all phase space points, there exists a negativity‐to‐violation map between Wigner function and quantum contextuality inequality. The robustness of the equivalence under depolarizing noise has been analyzed and discussed. For demonstration purposes, the graphs with different independence numbers and the corresponding set of states have been established on a single qutrit. Different to the cited work, this method involves only a single qudit, then is experimentally feasible for a qutrit.

Translations and reflections on the torus: identities for discrete Wigner functions and transforms

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.

State-injection schemes of quantum computation in Spekkens' toy theory

Spekkens' toy theory is a non-contextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. It has been shown in [3] that for qudits of odd dimensions it is operationally equivalent to stabiliser quantum mechanics by making use of Gross' theory of discrete Wigner functions. This result does not hold in the case of qubits, because of the unavoidable negativity of any Wigner function representation of qubit stabiliser quantum mechanics. In this work we define and characterise the subtheories of Spekkens' theory that are operationally equivalent to subtheories of stabiliser quantum mechanics. We use these Spekkens' subtheories as a unifying framework for the known examples of state-injection schemes where contextuality is an injected resource to reach universal quantum computation. In addition, we prove that, in the case of qubits, stabiliser quantum mechanics can be reduced to a Spekkens' subtheory in the sense that all its objects that do not belong to the Spekkens' subtheory, namely non-covariant Clifford gates, can be injected. This shows that within Spekkens' subtheories we possess the toolbox to perform state-injection of every object outside of them and it suggests that there is no need to use bigger subtheories to reach universal quantum computation via state-injection. We conclude with a novel scheme of computation suggested by our approach which is based on the injection of CCZ states and we also relate different proofs of contextuality to different state injections of non-covariant gates.

Multiqubit Clifford groups are unitary 3-designs

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.