Pauli Matrices Research Articles

Tailoring Fusion-Based Error Correction for High Thresholds to Biased Fusion Failures.

We introduce fault-tolerant (FT) architectures for error correction with the XZZX cluster state based on performing measurements of two-qubit Pauli operators Z⊗Z and X⊗X, or fusions, on a collection of few-body entangled resource states. Our construction is tailored to effectively correct noise that predominantly causes faulty X⊗X measurements during fusions. This feature offers a practical advantage in linear optical quantum computing with dual-rail photonic qubits, where failed fusions only erase X⊗X measurement outcomes. By applying our construction to this platform, we find a record-high threshold to fusion failures exceeding 25% in the experimentally relevant regime of nonzero loss rate per photon, considerably simplifying hardware requirements.

Growth Reduction of Similarity-Transformed Electronic Hamiltonians in Qubit Space.

Accurately solving the electronic structure problem through the variational quantum eigensolver (VQE) is hindered by the available quantum resources of current and near-term devices. One approach to relieving the circuit depth requirements for VQE is to "pre-process" the electronic Hamiltonian by a similarity transformation incorporating some degree of electronic correlation, with the remaining correlation left to be addressed by the circuit ansatz. This often comes at the price of a substantial increase in the number of terms to measure in the similarity-transformed Hamiltonian. In this work, we propose an efficient approach to sampling elements from the complete Pauli group for N qubits which minimizes the onset of new terms in the transformed Hamiltonian while facilitating substantial energy lowering. We benchmark the growth-mitigating generator selection technique for ground state energy estimations applied to models of the H4, N2, and H2O molecular systems. It is found that utilizing a selection procedure which obtains the growth-minimizing generator from the set of operators with the maximal energy gradient is the most competitive approach to reducing the onset of Hamiltonian terms while achieving systematic energy lowering of the reference state.

Resonances at the Threshold for Pauli Operators in Dimension Two

Resonances at the Threshold for Pauli Operators in Dimension Two

Hamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physics

We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a \(n\)-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval \(0.1\!-\!2\:\!\text{K}\), which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.

On the Pauli group on 2-qubits in dynamical systems with pseudofermions

Abstract The group of matrices P 1 {P_{1}} of Pauli is a finite 2-group of order 16 and plays a fundamental role in quantum information theory, since it is related to the quantum information on the 1-qubit. Here we show that both P 1 {P_{1}} and the Pauli 2-group P 2 {P_{2}} of order 64 on 2-qubits, other than in quantum computing, can also appear in dynamical systems which are described by non-self-adjoint Hamiltonians. This will allow us to represent P 1 {P_{1}} and P 2 {P_{2}} in terms of pseudofermionic operators.

The XZ Model in the Mean-Field Approximation with Exponential Pauli Spin Matrices

In this study, the mean-field approximation (MFA) is applied to the study of the XZ model for spin-1/2 in terms of Pauli spin matrices and their exponentials to obtain the magnetization components for the ferromagnetic (FM) and antiferromagnetic (AFM) cases. The details of obtaining the series expansion for the exponential Pauli spin matrices by using their fundamental properties and commutativity relations are presented. The partition function, free energy, and magnetization are obtained for a given coordination number in a simple form as hyperbolic tangent functions. The thermal variations of the magnetization components in the x- and z-directions are studied to obtain the phase diagrams in possible planes, which only present the second-order phase transitions. The reentrant behavior is also observed in the AFM case.

Violation of Bell Inequality by Four‐Photon Greenberger‐Horne‐Zeilinger State with a Phase from a Warm Atomic Ensemble

AbstractA Greenberger–Horne–Zeilinger (GHZ) entangled state with a phase is crucial for realizing desired multipartite quantum states for practical applications. Here, it is reported that the violations of the general Bell inequality (GBI) introduced in using the four‐photon polarization‐entangled phase‐GHZ state realized via intrinsic polarization correlation and collective two‐photon coherence in the 5S1/2–5P3/2–5D5/2 transition of 87Rb atoms. The phase‐GHZ state can be achieved by the unitary transformation of only one local phase of the four photons. Theoretically, the GHZ state with the π/4 phase affords maximal violation of the GBI of 2√2 at the local measurement settings of the Pauli operators σx and σy. Strong violations of the GBI of the phase‐GHZ state by 47 standard deviations are experimentally demonstrated. In addition to the entanglement witness for the phase‐GHZ state, the results represent a genuine four‐photon entanglement of the phase‐GHZ state, thereby providing a novel resource for realizing photonic quantum computation, magic‐state distillation from entangled states, and quantum networks based on atom–photon interactions.

An analytical solution for a Spin-1 Hamiltonian in the adiabatic approximation

In this work we present a complete analytic solution for a Hamiltonian that represents a three-level system described in terms of the spin 1 matrices, Jˆx and Jˆz, in interaction with a bosonic field written in terms of creation and annihilation operators, in the adiabatic approximation. This model is an immediate extension of the Quantum Rabi Model since we start from a Hamiltonian with the same structure but we replace the Pauli matrices of spin 1/2 by the Jˆ matrices of spin 1. We follow a procedure presented by Li and Batchelor in 2021 applied to a two-level system, extending it to a three-level system. In this case, the Hamiltonian is studied by blocks from two different perspectives: nine blocks of dimension n×n, where it is possible to identify displaced harmonic oscillators whose non-simultaneous solutions help us to build an eigenbase for Hˆ, and n2 blocks of dimension 3 × 3 which, in the adiabatic approximation regime, are reduced to n blocks that can be analytically diagonalized in a simple way. This allows us to generate all the eigenenergies of the Hamiltonian and their respective eigenvectors.

Detection of genuine multipartite entanglement in arbitrary multipartite systems

We study the genuine multipartite entanglement of arbitrary n-partite quantum states by representing the density matrices in terms of the generalized Pauli operators. We introduce a general framework for detecting genuine multipartite entanglement and non full-separability of multipartite quantum states with arbitrary dimensions based on correlation tensors. Effective criterion is derived to verify the genuine multipartite entanglement. Detailed examples are given to show that the criterion detects more genuine multipartite entanglement than the existing criteria.

The Modal Components of Judgements in a Quantum Model of Psychoanalytic Theory.

In the present paper, we develop a theory of thinking based on an attempt to formalize the construction of mental representations as described in psychoanalytic theory. In previous work, we described Freud's and Matte Blanco's structural Unconscious in a formal model in which the properties of unconscious representations are captured by particular sets-infinite singletons-that can be derived in first-order logic language. Here, we afford the issue of the finitization of unconscious representations by assuming that the mind can form an all-purpose modality, originating from abstraction from infinite singletons; in this way, a symmetric prelogical setting for mental representations is formally created, and this is interpreted in a quantum spin model by a modal (necessity) projector. Then, by introducing time, one can describe the links that mental representations can establish with reality, and hence finitize the representations. The modality is so split into finite components, here termed positive, negative and irreal; the splitting of the modality is traced back to the decomposition of the spin observables by means of the Pauli matrices, which can offer a quantum semantics to the method applied. Here, we suggest that the development of the modal approach and its quantum logic implementation can be considered as a proper formalization of some aspect of the psychoanalytic theory of thinking proposed by Bion; namely, we will show that the process of abstraction leading from raw data to preconceptions, and therefore to the definition of the content-container relationship, is adequately captured by our model, and further correspondences can be detected with Bion's theory about links and transformations, implying different ways in which the mind can get in touch with both internal and external reality.

Dirac equation from stereographic projection of the momentum sphere

The Dirac equation is commonly demonstrated under stringent hypotheses and after considerable math work made in relativistic quantum mechanics and quantum field theory. Here, a purely geometric approach free from hypotheses is suggested. The suggestion draws inspiration from the technique of stereographic projection that was developed before the quantum era to solve gyroscopic problems of classical mechanics. The projected variable is the generalized (or canonical) momentum vector. Its undetermined geometric orientations define a sphere in the momentum space and the projection onto the equatorial plane generates the Pauli matrices as soon as the conventional stereographic matrix is introduced. The associated eigenvalue problem results in the Dirac equation and the eigenvector (bispinor) has components that are related to geometric elements of the momentum space. The procedure has the advantage of revealing the correct form of the Dirac matrices without the mathematical effort that characterizes the presentation of the equation in traditional approaches. The other remarkable advantage is that, unlike the common reduction to the case of free space, the spatial inhomogeneity due to interaction potentials is included in the demonstration from the very beginning. The whole suggestion has an interdisciplinary character (relativity, complex analysis, rotation of rigid bodies, Pauli matrices) and can be useful in teaching the equation to students who lack in sufficient knowledge of quantum mechanics. Students equipped with more advanced education can benefit from the purely geometric perspective of this work if used to complement their studies about the equation.

Generalized Quaternions and Matrix Algebra

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
 Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

Efficient quantum secret sharing scheme based on monotone span program

How to efficiently share secrets among multiple participants is a very important problem in key management. In this paper, we propose a multi-secret sharing scheme based on the Greenberger Horne Zeilinger (GHZ) state. First, the distributor uses monotone span program to encode the secrets and generate the corresponding secret shares to send to the participants. Then, each participant uses the generalized Pauli operator to embed its own secret share into the transmitted particle. The participant who wants to get the secrets can get multiple secrets at the same time by performing a GHZ-state joint measurement. Furthermore, since this scheme is based on a monotone span program, its access structure is more general than the access structure (t, n) threshold. Compared with other schemes, our proposed scheme is more efficient, less computational cost.

Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers

Transformations which convert between fermionic modes and qubit operations have become a ubiquitous tool in quantum algorithms for simulating systems. Similarly, collections of Pauli operators might be obtained from solutions of nonlocal games and satisfiability problems. Drawing on ideas from entanglement-assisted quantum error-correcting codes and quantum convolutional codes, we prove the obtainable lower bound for the number of qubits needed to represent such Pauli operations which are equivalent to the initial ones and provide a procedure for determining such a set of minimal register Pauli operations.

Efficient Quantum Imaginary Time Evolution by Drifting Real-Time Evolution: An Approach with Low Gate and Measurement Complexity.

Quantum imaginary time evolution (QITE) is one of the promising candidates for finding the eigenvalues and eigenstates of a Hamiltonian on a quantum computer. However, the original proposal suffers from large circuit depth and measurements due to the size of the Pauli operator pool and Trotterization. To alleviate the requirement for deep circuits, we propose a time-dependent drifting scheme inspired by the qDRIFT algorithm [Campbell, E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme removes the depth dependency on the size of the operator pool and converges inversely with respect to the number of steps. We further propose a deterministic algorithm that selects the dominant Pauli term to reduce the fluctuation for the ground state preparation. We also introduce an efficient measurement reduction scheme across Trotter steps that removes its cost dependence on the number of iterations. We analyze the main source of error for our scheme both theoretically and numerically. We numerically test the validity of depth reduction, convergence performance of our algorithms, and the faithfulness of the approximation for our measurement reduction scheme on several benchmark molecules. In particular, the results on the LiH molecule give circuit depths comparable to that of the advanced adaptive variational quantum eigensolver (VQE) methods while requiring much fewer measurements.

Operator Relaxation and the Optimal Depth of Classical Shadows.

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via "shallow shadows," a recently proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth t. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing ("twirling") circuit-namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially contiguous Pauli operators of weight k, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i)an upper bound on the shadow norm which, for depth t∼log(k), guarantees an exponential gain in sample complexity over the t=0 protocol in any spatial dimension, and (ii)quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our Letter connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly optimized protocols for learning different properties of quantum states.

Scalable neural quantum states architecture for quantum chemistry

Variational optimization of neural-network representations of quantum states has been successfully applied to solve interacting fermionic problems. Despite rapid developments, significant scalability challenges arise when considering molecules of large scale, which correspond to non-locally interacting quantum spin Hamiltonians consisting of sums of thousands or even millions of Pauli operators. In this work, we introduce scalable parallelization strategies to improve neural-network-based variational quantum Monte Carlo calculations for ab-initio quantum chemistry applications. We establish GPU-supported local energy parallelism to compute the optimization objective for Hamiltonians of potentially complex molecules. Using autoregressive sampling techniques, we demonstrate systematic improvement in wall-clock timings required to achieve coupled cluster with up to double excitations baseline target energies. The performance is further enhanced by accommodating the structure of resultant spin Hamiltonians into the autoregressive sampling ordering. The algorithm achieves promising performance in comparison with the classical approximate methods and exhibits both running time and scalability advantages over existing neural-network based methods.

Quantum Secret Permutating Protocol

In modern cryptography, distributing a private and unique index number to each participant is an important cryptographic task, which can be adopted to efficiently solve many complicated secure multiparty computations. In this paper, we define this cryptographic primitive, called Secret Permutating, in which every one of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bm{n}}$</tex-math></inline-formula> participants can get a random but unique secret <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\bm{k}}}_{\bm{i}} \in \{ {1,2, \ldots ,{\bm{n}}} \}$</tex-math></inline-formula> . Furthermore, we focus on the unconditional security of Secret Permutating based on laws of quantum mechanics. Accordingly, by local Pauli operators and entanglement swapping of Bell states, we design novel quantum Secret Permutating protocols. What's more, to reduce the communicational complexity, we exploit the uniform, random and independent properties of quantum measurements to evenly divide all participants into many secret groups with the small approximate sizes. Finally, the analysis results and simulated experiments show that the proposed protocols have the unconditional security and the good feasibility.

Lower Bound for the T Count Via Unitary Stabilizer Nullity

We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on the $T$ count for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverland et al. to a dynamic version. In particular, we show this nonstabilizerness measure has desirable properties such as subadditivity under composition and additivity under tensor product. Second, we prove that a given unitary's stabilizer nullity is a lower bound for the $T$ count, utilizing the above properties in gate synthesis. Third, we compare the state- and the unitary-stabilizer nullity, proving that the lower bounds for the $T$ count obtained by the unitary-stabilizer nullity are never less than the state-stabilizer nullity. Moreover, we show an explicit $n$-qubit unitary family of unitary-stabilizer nullity $2n$, which implies that its $T$ count is at least $2n$. This gives an example where the bounds derived by the unitary-stabilizer nullity strictly outperform the state-stabilizer nullity by a factor of $2$. We finally showcase the advantages of unitary-stabilizer nullity in estimating the $T$ count of quantum gates with interests.

Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions

We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Chen et al. [Ann. Phys. (N. Y) 393, 234 (2018)], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice. We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios r=1+1/2k for positive integers k, utilizing the trivialness of quantum cellular automata in two spatial dimensions. Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1. When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev’s exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.21 MoreReceived 5 October 2022Revised 30 January 2023Accepted 16 February 2023DOI:https://doi.org/10.1103/PRXQuantum.4.010326Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBosonizationFermionsQuantum error correctionQuantum simulationTechniquesDualityQuantum Information