Pauli Matrices Research Articles

Unconditional quantum magic advantage in shallow circuit computation

Quantum theory promises computational speed-ups over classical approaches. The celebrated Gottesman-Knill Theorem implies that the full power of quantum computation resides in the specific resource of “magic” states—the secret sauce to establish universal quantum computation. However, it is still questionable whether magic indeed brings the believed quantum advantage, ridding unproven complexity assumptions or black-box oracles. In this work, we demonstrate the first unconditional magic advantage: a separation between the power of generic constant-depth or shallow quantum circuits and magic-free counterparts. For this purpose, we link the shallow circuit computation with the strongest form of quantum nonlocality—quantum pseudo-telepathy, where distant non-communicating observers generate perfectly synchronous statistics. We prove quantum magic is indispensable for such correlated statistics in a specific nonlocal game inspired by the linear binary constraint system. Then, we translate generating quantum pseudo-telepathy into computational tasks, where magic is necessary for a shallow circuit to meet the target. As a by-product, we provide an efficient algorithm to solve a general linear binary constraint system over the Pauli group, in contrast to the broad undecidability in constraint systems. We anticipate our results will enlighten the final establishment of the unconditional advantage of universal quantum computation.

Hardware-tailored diagonalization circuits

A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable SWAP gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and – as a proof-of-principle application – we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers.

Quantum Interpretation for Measurements in Physics

The postulate in the Copenhagen interpretation is for quantum measures that in experiments from a set of operators only a subset of commuting operators can be measured simultaneously, the rest remains undetermined. As example the Stern-Gerlach measure shows for spin a 90 degree change and measures for instance spin up and down of the outcome particles only in the z-direction. The spacial xy-coordinates remain undetermined.Using octonion coordinates [1,2] for the quantum range this view can be applied to all octonion base GF triples which use the three noncommuting Pauli matrices. The coordinates are enumerated by indices 0,1,2,…,7. Entanglements of different GF are possible. Two examples are the spin-lepton cases 123 (for xyz) and 145 (1 for electrical EM charge, 4 for magnetism, 5 for leptonic mass) either in the gyromagnetic relation (EM) or the helicty (neutral leptons).A in the second case the Copenhagen interpretation is extended to a quantum interpretation for measurements of the GF. The systems and energies involved can be different. The neutrino N oscillations show that also the Heisenberg uncertainties HU play a role. Spin is aligned with the space coordinate 1 and the momentum p = mv on 6. The HU means that 1,6 cannot be measured sharp. Hence the leptonic kg measure 5 changes for p along the world line of N stochasticaly, observed as oscillation. The kg GF is 257 and has 6 possible values for leptons. The weights of the three GF coordinates are mostly positive real or complex numbers. There are seven octonion GF 123, 145, 167 (for electromagnetic interaction EMI), 246 (for heat, acoustics), 257, 347 (for rotational energy), 356 (for a nucleon inner dynamics). Beside these are for the strong interaction SI three more GF 126 as rgb-graviton , 345 for its dual Dg and 037 for a newly postulated color charge force cc.The new cc force has a different symmetry than the Pauli quaternions.The six complex cross ratios invariant under the Moebius transformations of the Riemannian sphere are discussed in relation to quantum measures.

Decoherence time control by interconnection for finite-level quantum memory systems

This paper is concerned with open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices for finite-level systems. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics according to a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This setting includes the zero-Hamiltonian isolated system dynamics as a particular case, where the system variables are constant in time, which makes them potentially applicable as a quantum memory. In a more realistic case of nonvanishing system-field coupling, we define a memory decoherence time when a mean-square deviation of the system variables from their initial values becomes relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the decoherence time maximization over the energy parameters of the system and obtain a condition under which the zero Hamiltonian provides a suboptimal solution. This optimization problem is also discussed for a direct energy coupling interconnection of such systems.

Qudit stabilizer codes, CFTs, and topological surfaces

We study general maps from the space of rational conformal field theories (CFTs) with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full Abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces. Published by the American Physical Society 2024

A new matrix representation of the Maxwell equations based on the Riemann–Silberstein–Weber vector for a linear inhomogeneous medium

We derive a new eight dimensional matrix representation of the Maxwell equations for a linear homogeneous medium and extend it to the case of a linear inhomogneous medium. This derivation starts ab initio with the Maxwell equations and uses arguments based on the algebra of the Pauli matrices. This process leads automatically to the matrix representation based on the Riemann–Silberstein–Weber (RSW) vector. The new representation for the homogeneous medium is a direct sum of four Pauli matrix blocks. This aspect of the new representation should make it suitable for studying the propagation of electromagnetic waves in a linear inhomogeneous medium adopting the techniques of quantum mechanics treating the inhomogeneity as a perturbation. The new representation is used to rederive the Mukunda–Simon–Sudarshan matrix substitution rule for transition from the Helmholtz scalar wave optics to the Maxwell vector wave optics.

Efficient multi-party quantum secret-sharing protocol

ContextQuantum secret sharing allows one secret dealer to divide his/her secret among different participants. Only when all the participants pool their secret pieces together can the secret be reconstructed. Quantum secret-sharing protocol can provide high security for users who need to share some secret information among some partners. It is also a useful tool in many applications such as secure operations of joint sharing of quantum money, sharing ancilla states, and distributed quantum computation. ObjectiveWe propose a novel multi-party quantum secret-sharing protocol (MPQSSP). MethodsIn our protocol, the dealer mixes the particle sequence of the Bell states and sends the sequence to the participants. The participants encode their secret pieces into Pauli operators, which are applied to the received particle sequence. After annularly transmitting and operating the particle sequence, the participants return the particle sequence to the dealer, who can share a secret with the partners by measuring the returned sequence with the Bell basis. ResultsThe proposed MPQSSP has the merits as follows. First, its qubit efficiency can be 100% when the sample bits of the secret and the corresponding sample qubits are ignored (In our protocol when the qubit efficiency is computed, the sample bits and the corresponding Bell states used to check eavesdropping are ignored). Second, it can protect participants' privacy and interests, since the dealer cannot derive the secret piece of any participant although he/she masters the secret. Third, the participants need not perform many quantum operations and measurements to check the eavesdropping, which helps improve the efficiency of the protocol. Fourth, compared with most of the similar MPQSSPs, the quantum resources of the proposed one are relatively easier to prepare. At last, the proposed protocol has the security against various eavesdropping attacks. ConclusionThe merits above show that Our MPQSSP is relatively more practical and efficient than similar ones.

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan–Wigner form in which they can act on multi-qubit states. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations for many-body systems.

Quantitative Hardy inequalities for magnetic Hamiltonians

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.

Optimal Twirling Depth for Classical Shadows in the Presence of Noise.

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state ρ using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i)show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii)compute thresholds f_{th} at which optimal twirling reduces to local twirling for Pauli operators, (iii)nth order Renyi entropies (n≥2), and (iv)provide a meaningful upper bound t_{max} on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.

Improving bidirectional controlled remote preparation of arbitrary number of qudits in noisy environment

By elaborately selecting entangled channel, we put forward a universal scheme to achieve bidirectional remote preparation of qudit states with arbitrary numbers of particles under the control of the supervisor. Each sender needs to perform a positive-operator-value measurement and a projective measurement, while the supervisor requires to execute a projective measurement. Based on their measurement outcomes, the receivers can recover the desired states concurrently by carrying out single-qudit generalized Pauli operations which are given by general formulas. A point to highlight is that classical communication cost is greatly reduced at the supervisor’s broadcast channel by taking advantage of network coding. In addition, we analyze the influence of amplitude-damping and phase-damping noises, and utilize weak measurement and environment-assisted measurement to suppress noise.

Topological phases of tight-binding trimer lattice in the BDI symmetry class

In this work, we theoretically study a modified Su-Schrieffer-Heeger (SSH) model in which each unit cell consists of three sites. Unlike existing extensions of the SSH model which are made by enlarging the periodicity of the (nearest-neighbor) hopping amplitudes, our modification is obtained by replacing the Pauli matrices in the system's Hamiltonian by their higher dimensional counterparts. This, in turn, leads to the presence of next-nearest neighbor hopping terms and the emergence of different symmetries than those of other extended SSH models. Moreover, the system supports a number of edge states that are protected by a combination of particle-hole, time-reversal, and chiral symmetry. Finally, our system could be potentially realized in various experimental platforms including superconducting circuits as well as acoustic/optical waveguide arrays.

An exponential reduction in training data sizes for machine learning derived entanglement witnesses

Abstract We propose a support vector machine (SVM) based approach for generating an entanglement witness that requires exponentially less training data than previously proposed methods. SVMs generate hyperplanes represented by a weighted sum of expectation values of local observables whose coefficients are optimized to sum to a positive number for all separable states and a negative number for as many entangled states as possible near a specific target state. Previous SVM-based approaches for entanglement witness generation used large amounts of randomly generated separable states to perform training, a task with considerable computational overhead. Here, we propose a method for orienting the witness hyperplane using only the significantly smaller set of states consisting of the eigenstates of the generalized Pauli matrices and a set of entangled states near the target entangled states. With the orientation of the witness hyperplane set by the SVM, we tune the plane's placement using a differential program that ensures perfect classification accuracy on a limited test set as well as maximal noise tolerance. For \(N\) qubits, the SVM portion of this approach requires only \(O(6^N)\) training states, whereas an existing method needs \(O(2^{4^N})\). We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 percent and 1 percent, respectively. We also use the same training states to generate novel 4 and 5 qubit W state witnesses. Finally, we computationally verify these witnesses on small test sets and propose methods for further verification.

X, Y, and Z Subgroups of the Unitary Group

The subgroups XU(2), YU(2), and ZU(2) of the unitary group U(2) allow to find twenty-four different decompositions of an arbitrary [Formula: see text] unitary matrix. Because the group YU(2) has not been considered before, 20 out of the 24 decompositions are new. Introducing this YU(2) group allows to highlight, within quantum circuit design, a symmetry similar to the threefold symmetry of the Pauli matrices of spin systems. Similar subgroups of the unitary group U([Formula: see text]) lead to various new decompositions of an arbitrary [Formula: see text] unitary matrix. Whereas for [Formula: see text] equal 2, this leads to the synthesis of single-qubit quantum gates, for [Formula: see text] equal a power of 2, this leads to new design methods for multiple-qubit circuits.

Quantum annealer accelerates the variational quantum eigensolver in a triple-hybrid algorithm

Hybrid algorithms that combine quantum and classical resources have become commonplace in quantum computing. The variational quantum eigensolver (VQE) is routinely used to solve prototype problems. Currently, hybrid algorithms use no more than one kind of quantum computer connected to a classical computer. In this work, a novel triple-hybrid algorithm combines the effective use of a classical computer, a gate-based quantum computer, and a quantum annealer. The solution of a graph coloring problem found using a quantum annealer reduces the resources needed from a gate-based quantum computer to accelerate VQE by allowing simultaneous measurements within commuting groups of Pauli operators. We experimentally validate our algorithm by evaluating the ground state energy of H2 using different IBM Q devices and the DWave Advantage system requiring only half the resources of standard VQE. Other larger problems we consider exhibit even more significant VQE acceleration. Several examples of algorithms are provided to further motivate a new field of multi-hybrid algorithms that leverage different kinds of quantum computers to gain performance improvements.

Fully Scalable Randomized Benchmarking Without Motion Reversal

We introduce , a protocol that streamlines traditional RB by using circuits consisting almost entirely of independent identically distributed (IID) layers of gates. BiRB reliably and efficiently extracts the average error rate of a Clifford gate set by sending tensor-product eigenstates of random Pauli operators through random circuits with IID layers. Unlike existing RB methods, BiRB does not use motion reversal circuits—i.e., circuits that implement the identity (or a Pauli) operator—which simplifies both the method and the theory proving its reliability. Furthermore, this simplicity enables scaling BiRB to many more qubits than the most widely used RB methods. Published by the American Physical Society 2024

Multiparty Quantum Key Agreement Based on d$d$‐dimensional Bell States

AbstractIn this paper, on the design process of a multiparty quantum key agreement protocol within a ‐dimensional Hilbert space is elaborated upon. A circular‐type multiparty quantum key agreement protocol based on the generalized Bell state is introduced. To enhance security against external attacks, decoy states into the transmission process is incorporated. Transmission sequences of the generalized Bell state and decoy states are passed between participants. The participants then encode their secret information into the corresponding particles. Ultimately, all participants are able to derive the same key. In addition, a combination of ‐dimensional Pauli operators is utilized, making the proposed protocol feasible with current technology. Analysis and protection against intercept‐resend attack, entangle‐measure attack and dishonest participants attacks, demonstrating the feasibility of the protocol in a ‐dimensional Hilbert space. The protocol has certain advantages over other protocols in terms of a comprehensive consideration of security and efficiency.

Efficient quantum state estimation with low-rank matrix completion

This paper introduces a novel and efficient technique for quantum state estimation, coined as low-rank matrix-completion quantum state tomography for characterizing pure quantum states, as it requires only non-entangling bases and 2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2n + 1$\\end{document} local Pauli operators. This significantly reduces the complexity of the process and increases the accuracy of the state estimation, as it eliminates the need for the entangling bases, which are experimentally difficult to implement on quantum devices. The required minimal post-processing, improved accuracy and efficacy of this matrix-completion-based method make it an ideal benchmarking tool for investigating the properties of quantum systems, enabling researchers to verify the accuracy of quantum devices, characterize their performance, and explore the underlying physics of quantum phenomena. Our numerical results demonstrate that this method outperforms contemporary techniques in its ability to accurately reconstruct multi-qubit quantum states on real quantum devices, making it an invaluable contribution to the field of quantum state characterization and an essential step toward the reliable deployment of intermediate- and large-scale quantum devices.

Absorbing Boundary Conditions Derived Based on Pauli Matrices Algebra

Absorbing Boundary Conditions Derived Based on Pauli Matrices Algebra

Detecting Entanglement from Macroscopic Measurements of the Electric Field and Its Fluctuations.

To address the outstanding task of detecting entanglement in large quantum systems, entanglement witnesses have emerged, addressing the separable nature of a state. Yet optimizing witnesses, or accessing them experimentally, often remains a challenge. We here introduce a family of entanglement witnesses for open quantum systems. Based on the electric field, it does not require state tomography or single-site addressing, but rather macroscopic measurements of the field quadratures and of the total fluorescence. Its efficiency is demonstrated by detecting, from almost any direction, the entanglement of collective single-photon states, such as long-lived states generated by cooperative spontaneous emission. Able to detect entanglement in large open quantum systems, and through a single continuous measurement if operating in the stationary regime, these electric-field-based witnesses can be used on any set of emitters described by the Pauli group, such as atomic systems (cold atoms and trapped ions), giant atoms, color centers, and superconducting qubits.