Pauli Matrices Research Articles

Sobre grupos clásicos de la física matemática

Starting from Pauli and Dirac matrices of 1928 we present a friendly and unified version of the classical groups of mathematical physics as subgroups of sub algebras of real geometric algebras, created and presented for Clifford in 1879, the prior concept of Clifford algebras.

Un-Weyl-ing the Clifford Hierarchy

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

Relevant out-of-time-order correlator operators: Footprints of the classical dynamics.

The out-of-time-order correlator (OTOC) has recently become relevant in different areas where it has been linked to scrambling of quantum information and entanglement. It has also been proposed as a good indicator of quantum complexity. In this sense, the OTOC-RE theorem relates the OTOCs summed over a complete basis of operators to the second Renyi entropy. Here we have studied the OTOC-RE correspondence on physically meaningful bases like the ones constructed with the Pauli, reflection, and translation operators. The evolution is given by a paradigmatic bi-partite system consisting of two perturbed and coupled Arnold cat maps with different dynamics. We show that the sum over a small set of relevant operators is enough in order to obtain a very good approximation for the entropy and, hence, to reveal the character of the dynamics. In turn, this provides us with an alternative natural indicator of complexity, i.e., the scaling of the number of relevant operators with time. When represented in phase space, each one of these sets reveals the classical dynamical footprints with different depth according to the chosen basis.

Enhanced noise resilience of the surface–Gottesman-Kitaev-Preskill code via designed bias

We study the code obtained by concatenating the standard single-mode Gottesman-Kitaev-Preskill (GKP) code with the surface code. We show that the noise tolerance of this surface-GKP code with respect to (Gaussian) displacement errors improves when a single-mode squeezing unitary is applied to each mode assuming that the identification of quadratures with logical Pauli operators is suitably modified. We observe noise-tolerance thresholds of up to $\sigma\approx 0.58$ shift-error standard deviation when the surface code is decoded without using GKP syndrome information. In contrast, prior results by Fukui et al. and Vuillot et al. report a threshold between $\sigma\approx 0.54$ and $\sigma\approx 0.55$ for the standard (toric-, respectively) surface-GKP code. The modified surface-GKP code effectively renders the mode-level physical noise asymmetric, biasing the logical-level noise on the GKP-qubits. The code can thus benefit from the resilience of the surface code against biased noise. We use the approximate maximum likelihood decoding algorithm of Bravyi et al. to obtain our threshold estimates. Throughout, we consider an idealized scenario where measurements are noiseless and GKP states are ideal. Our work demonstrates that Gaussian encodings of individual modes can enhance concatenated codes.

Extended flag gadgets for low-overhead circuit verification

Flag verification techniques are useful in quantum error correction for detecting critical faults. Here we present an application of flag verification techniques to improving post-selected performance of near-term algorithms. We extend the definition of what constitutes a flag by creating error-detection gadgets based on known transformations of unitary operators. In the case of Clifford or near-Clifford circuits, these unitary operators can be chosen to be controlled Pauli gates, leading to gadgets which require only a small number of additional Clifford gates. We show that such flags can improve circuit fidelities by up to a factor of 2 after post selection, and demonstrate their effectiveness over error models featuring single-qubit depolarizing noise, crosstalk, and two-qubit coherent overrotation.

Pauli matrices immersion

Spin polarization, a major consideration in materials science, has a three-dimensional Euclidean background, but quantum spin has a non-Euclidean ambient space of the Dirac spinor. This paper bridges the gap by the linear algebra of complexification to cast imaginary numbers in a two-dimensional real space. As real and imaginary eigenvalues together describe motions in the physical space, we immerse Pauli matrices in a three-dimensional ambient space. Thereupon we present an alternative explanation of the anti-matter asymmetry, with implications on spin polarization in electronic engineering and materials science.

Pauli Matrices: A Triple of Accardi Complementary Observables

The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $ su(2) $. We show that the pair of Pauli matrices $ A,B $ associated to the unit directions $ \alpha $ and $ \beta $ in $ \mathbb{R}^{3} $ are Accardi complementary if and only if $ \alpha $ and $ \beta $ are orthogonal if and only if $ A $ and $ B $ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary.

Electronic Maxwell’s equations

To date, the wave nature of electron has been widely researched, together with its similarity to optics. To unify electronic waves and electromagnetic waves, we establish four equations analogous to Maxwell’s equations by expressing the Dirac equation in terms of the quaternions. We develop some fundamental theories from the electronic version of Maxwell’s equations. In practice, solving electron wave problem is convenient without appearance of quantum operators such as Pauli matrices. The benefit is its potential in the analysis and applications of diverse electron beams, for example, phase-shift control. Moreover, physical quantities represented by electronic vector fields are notably similar to those in optics, making it possible to apply some ideas developed in optics in the research of electron.

FeynOnium: using FeynCalc for automatic calculations in Nonrelativistic Effective Field Theories

We present new results on FeynOnium, an ongoing project to develop a general purpose software toolkit for semi-automatic symbolic calculations in nonrelativistic Effective Field Theories (EFTs). Building upon FeynCalc, an existing Mathematica package for symbolic evaluation of Feynman diagrams, we have created a powerful framework for automatizing calculations in nonrelativistic EFTs (NREFTs) at tree- and 1-loop level. This is achieved by exploiting the novel features of FeynCalc that support manipulations of Cartesian tensors, Pauli matrices and nonstandard loop integrals. Additional operations that are common in nonrelativistic EFT calculations are implemented in a dedicated add-on called FeynOnium. While our current focus is on EFTs for strong interactions of heavy quarks, extensions to other systems that admit a nonrelativistic EFT description are planned for the future. All our codes are open-source and publicly available. Furthermore, we provide several example calculations that demonstrate how FeynOnium can be employed to reproduce known results from the literature.

Predicting excited states from ground state wavefunction by supervised quantum machine learning

Excited states of molecules lie in the heart of photochemistry and chemical reactions. The recent development in quantum computational chemistry leads to inventions of a variety of algorithms that calculate the excited states of molecules on near-term quantum computers, but they require more computational burdens than the algorithms for calculating the ground states. In this study, we propose a scheme of supervised quantum machine learning which predicts the excited-state properties of molecules only from their ground state wavefunction resulting in reducing the computational cost for calculating the excited states. Our model is comprised of a quantum reservoir and a classical machine learning unit which processes the measurement results of single-qubit Pauli operators with the output state from the reservoir. The quantum reservoir effectively transforms the single-qubit operators into complicated multi-qubit ones which contain essential information of the system, so that the classical machine learning unit may decode them appropriately. The number of runs for quantum computers is saved by training only the classical machine learning unit, and the whole model requires modest resources of quantum hardware that may be implemented in current experiments. We illustrate the predictive ability of our model by numerical simulations for small molecules with and without noise inevitable in near-term quantum computers. The results show that our scheme reproduces well the first and second excitation energies as well as the transition dipole moment between the ground states and excited states only from the ground states as inputs. We expect our contribution will enhance the applications of quantum computers in the study of quantum chemistry and quantum materials.

An Improved Noise Quantum Annealing Method for TSP

Traveling Salesman Problem (TSP) is a combinatorial optimization problem, which has NP-Complete (NPC) complexity. At present, there is no exact algorithm to solve similar problems, only an approximate algorithm to simplify the problem can be found. For example, quantum annealing algorithm (QA) can play such a role. QA transforms the distance matrix sum of TSP into Pauli matrix, adds a transverse magnetic field to realize the quantum tunneling effect, reduces the energy needed to cross the barrier, and reduces the number of iterations to find the optimal solution. However, although the QA has fewer iteration steps, it usually produces errors. In this context, this paper’s INQA improves the QA. In this paper, the theoretical basis of quantum annealing algorithm is introduced, aiming at the shortcomings of quantum annealing algorithm in solving TSP problem, a new algorithm INQA is proposed, and it is verified by experiments. In the case of high initial temperature, the INQA has larger search range and more active search. Meanwhile, with the decrease of temperature, the search range is reduced, and the accuracy of the algorithm is improved. In fact, QA in this paper is a quantum annealing method for simulation.

Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.

Complementary properties of multiphoton quantum states in linear optics networks

We have developed a theory for accessing quantum coherences in mutually unbiased bases associated with generalized Pauli operators in multiphoton multimode linear optics networks (LONs). We show a way to construct complementary Pauli measurements in multiphoton LONs and establish a theory for evaluation of their photonic measurement statistics without dealing with the computational complexity of Boson samplings. This theory extends characterization of complementary properties in single-photon LONs to multiphoton LONs employing convex-roof extension. It allows us to detect quantum properties such as entanglement using complementary Pauli measurements, which reveals the physical significance of entanglement between modes in bipartite multiphoton LONs.

Spin s representations via sigma models* *The research of AMG and AMER was supported by the Natural Sciences and Engineering Research Council of Canada operating grant of one of the authors (AMG).

We establish and analyze a new relationship between the matrix functions describing spin fields of a spin s, where , and two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the anti-Hermitian immersion functions of their soliton surfaces in the algebra. We provide a geometric interpretation of this construction. For the spin fields which can be represented as linear combinations of the generalized Pauli matrices, we find the dynamics equation satisfied by their coefficients. This equation is identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for matrices congruent to the generalized Pauli matrices through any coordinate-independent unitary linear transformation. These properties allow for new interpretations of the spins as compositions of more elementary objects. They also open up the possibility of future applications of the sigma models to the situations which depend on spin behaviour, including spintronics, spin glasses and quantum computing.

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters

Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number ofCNOToperations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.

(t,n) Threshold d-level QSS based on QFT

Quantum secret sharing (QSS) is an important branch of secure multiparty quantum computation. Several schemes for (n, n) threshold QSS based on quantum Fourier transformation (QFT) have been proposed. Inspired by the flexibility of (t, n) threshold schemes, Song {\it et al.} (Scientific Reports, 2017) have proposed a (t, n) threshold QSS utilizing QFT. Later, Kao and Hwang (arXiv:1803.00216) have identified a {loophole} in the scheme but have not suggested any remedy. In this present study, we have proposed a (t, n)threshold QSS scheme to share a d dimensional classical secret. This scheme can be implemented using local operations (such as QFT, generalized Pauli operators and local measurement) and classical communication. Security of the proposed scheme is described against outsider and participants' eavesdropping.

High-Spin Paramagnetic Ions as Qubits and Qutrits for Quantum Computations

Paramagnetic ions having two electrons and spin S = 1 in zero magnetic field are proposed for new physical performance of elementary quantum computing operations by pulsed manipulations with electron spins. For this purpose, a new type of microwave pulses with variable phases and different polarizations are shown can be used. Such pulses are able to induce different transitions between ion spin states. Some of those transitions are described by operators similar to quantum computing operators. To describe the effects of such pulses on hig-spin ions, a new way to represent exponential operators as polynomials of the Pauli matrices is proposed.

Bidirectional teleportation using Fisher information

In this contribution, we reformulated the bidirectional teleportation protocol suggested in Ref. 7, by means of Bloch vectors as well as the local operations are represented by using Pauli operators. Analytical and numerical calculations for the teleported state and Fisher information are introduced. It is shown that both quantities depend on the initial state settings of the teleported qubits and their triggers. The Fidelities and the Fisher information of the bidirectionally teleported states are maximized when the qubit and its trigger are polarized in the same direction. The minimum values are predicted if both initial qubits have different polarization or nonzero phase. The maximum values of the Fidelity and the quantum Fisher information are the same, but they are predicted at different polarization angles. We display that the multi-parameter form is much better than the single parameter form, where it satisfies the bounds of classical, entangled systems and the uncertainty principle.

Rigidity and a common framework for mutually unbiased bases and k‐nets

AbstractMany deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called ‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a ‐net over a ‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) ‐nets, while the choice of leads to collections of MUBs. In this framework, we derive a rigidity property which hence automatically applies to both objects. For ‐nets that can be completed to affine planes, this was already known by a completely different, combinatorial argument. For ‐nets that cannot be completed and for MUBs, this result is new, and it implies that the only vectors unbiased to all but bases of a complete collection of MUBs in are the elements of the remaining bases (up to phase factors). Further, we show that this bound is tight with counterexamples for in every prime‐square dimension. Applying our rigidity result, we prove that if a large enough collection of MUBs constructed from a certain unitary error basis (like, the generalized Pauli operators) can be extended to a complete system, then every basis of the completion must come from the same error basis. In turn, we use this to show that certain large systems of MUBs cannot be completed.