A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

When visualised as an operation on the Bloch sphere, the qubit\n"pi-over-eight" gate corresponds to one-eighth of a complete rotation about the\nvertical axis. This simple gate often plays an important role in quantum\ninformation theory, typically in situations for which Pauli and Clifford gates\nare insufficient. Most notably, when it supplements the set of Clifford gates\nthen universal quantum computation can be achieved. The "pi-over-eight" gate is\nthe simplest example of an operation from the third level of the Clifford\nhierarchy (i.e., it maps Pauli operations to Clifford operations under\nconjugation). Here we derive explicit expressions for all qudit (d-level, where\nd is prime) versions of this gate and analyze the resulting group structure\nthat is generated by these diagonal gates. This group structure differs\ndepending on whether the dimensionality of the qudit is two, three or greater\nthan three. We then discuss the geometrical relationship of these gates (and\nassociated states) with respect to Clifford gates and stabilizer states. We\npresent evidence that these gates are maximally robust to depolarizing and\nphase damping noise, in complete analogy with the qubit case. Motivated by this\nand other similarities we conjecture that these gates could be useful for the\ntask of qudit magic-state distillation and, by extension, fault-tolerant\nquantum computing. Very recent, independent work by Campbell, Anwar and Browne\nconfirms the correctness of this intuition, and we build upon their work to\ncharacterize noise regimes for which noisy implementations of these gates can\n(or provably cannot) supplement Clifford gates to enable universal quantum\ncomputation.\n

It is proved that the braid group representation constructed in terms of the quantum group can be covered by that obtained from the proposal of weight conservation by the extended Kauffman diagram technique. The non-standard braid group representations associated with SU(2) are obtained by adding the terms of weight conservation to the standard universal R-matrix.

There is a property of a quantum state called ``magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: ``mana'' and ``robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes ``maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R\'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.

What is a quantum computer? It is tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behavior of all physical phenomena, this temptation must be resisted. Your laptop operates under the laws of quantum mechanics, but it is not a quantum computer. A quantum computer is one whose operation exploits certain very special transformations of its internal state, whose description is the primary subject of this book. The laws of quantum mechanics allow these peculiar transformations to take place under very carefully controlled conditions. In a quantum computer the physical systems that encode the individual logical bits must have no physical interactions whatever that are not under the complete control of the program. All other interactions, however irrelevant they might be in an ordinary computer – which we shall call classical – introduce potentially catastrophic disruptions into the operation of a quantum computer. Such damaging encounters can include interactions with the external environment, such as air molecules bouncing off the physical systems that represent bits, or the absorption of minute amounts of ambient radiant thermal energy. There can even be disruptive interactions between the computationally relevant features of the physical systems that represent bits and other features of those same systems that are associated with computationally irrelevant aspects of their internal structure.

The author studies the higher-order rewrite/equational proof systems obtained by adding the simply typed lambda calculus to algebraic rewrite/equational proof systems. He shows that if a many-sorted algebraic rewrite system has the Church-Rosser property, then the corresponding higher-order rewrite system which adds simply typed beta -reduction has the Church-Rossers property too. This result is relevant to parallel implementations of functional programming languages. The author also shows that provability in the higher-order equational proof system obtained by adding the simply typed beta and eta axions to some many-sorted algebraic proof system is effectively reducible to provability in that algebraic proof system. This effective reduction also establishes transformations between higher-order and algebraic equational proofs, which can be useful in automated deduction.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A frequency domain proof for the equivalence of the maximally flat FIR fractional delay filter and the Lagrange interpolator

Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schrödinger's dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind's idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)-representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

In this paper we describe connections among extraspecial 2-groups, unitary representa-tions of the braid group and multi-qubit braiding quantum gates. We first construct newrepresentations of extraspecial 2-groups. Extending the latter by the symmetric group,we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates gen-erate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly anodd) number of qubits, from the product basis. We also discuss the Yang-Baxterizationof the new braid group representations, which describes unitary evolution of the GHZstates. Our study suggests that through their connection with braiding gates, extraspe-cial 2-groups and the GHZ states may play an important role in quantum error correctionand topological quantum computing.

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.

In this paper, we present the experimental realization of multi-qubit gates $% \Lambda_n(not) $ in macroscopic ensemble of three-qubit and four-qubit molecules. Instead of depending heavily on the two-bit universal gate, which served as the basic quantum operation in quantum computing, we use pulses of well-defined frequency and length that simultaneously apply to all qubits in a quantum register. It appears that this method is experimentally convenient when this procedure is extended to more qubits on some quantum computation, and it can also be used in other physical systems.

Extensive experiments have demonstrated quantum behaviour in the long-time operation of the D-Wave quantum computer. The decoherence time of a single flux qubit is reported to be on the order of nanoseconds [1], which is much shorter than the time required to carry out a computation on the timescale of seconds [2, 3]. Previous judgements of whether the D-Wave device should be thought of as a quantum computer have been based on correlations of the input-output behaviour of the D-Wave machine with a quantum model, called simulated quantum annealing, or classical models, called simulated annealing and classical spin dynamics [4]. Explanations for a factor of 108 discrepancy between the single flux qubit decoherence time and the long-time coherent quantum behaviour of many integrated flux qubits of the D-Wave device have not been offered so far. In our contribution we investigate a model of four qubits with one qubit coupled to a phonon and (optionally) to environmental particles of high density of states, called gravonons. The calculations indicate that when no gravonons are present, the current in the qubit is flipped at some time and adiabatic evolution is discontinued. The time dependent wave functional becomes a non-correctable superposition of many excited states. The results demonstrate the possibility of effectively suppressing the current flip and allowing for continued adiabatic evolution when the entanglement to gravonons is included. This adiabatic evolution is, however, a coherent evolution in high dimensional spacetime and cannot be understood as a solution of Schrödinger's time dependent equation in 4 dimensional spacetime. Compared to Schrödinger's time development in 4D, the evolution is considerably slowed down, though still adiabatic. The properties of our model reflect correctly the experimentally found behaviour of the D-Wave machine and explain the factor of 108 discrepancy between decoherence time and quantum computation time. The observation and our explanation are in anology to the 108 discrepancy factor found, when comparing experimental results on adsorbate quantum diffusion rate with predictions of Schrödinger's time dependent equation, which can also be resolved in a model with the coupling to gravonons included.

An architecture for a quantum computer is presented in which spins associated\nwith donors in silicon function as qubits. Quantum operations on the spins are\nperformed using a combination of voltages applied to gates adjacent to the\nspins and radio frequency applied magnetic fields resonant with spin\ntransitions. Initialization and measurement of electron spins is made by\nelectrostatic probing of a two electron system, whose orbital configuration\nmust depend on the spin states of the electrons because of the Pauli Principle.\nSpecific devices will be discussed which perform all the necessary operations\nfor quantum computing, with an emphasis placed on the qualitative principles\nunderlying their operation.\n The likely impediments to achieving large-scale quantum computation using\nthis architecture will be addressed: the computer must operate at extremely low\ntemperature, must be fabricated from devices built with near atomic precision,\nand will require extremely accurate gating operations in order to perform\nquantum logic. Refinements to the computer architecture will be presented which\ncould remedy each of these deficiencies. I will conclude by discussing a\nspecific realization of the computer using Si/SiGe heterostructures into which\ndonors are deposited using a low energy focused ion beam.\n

Parallel operations in conventional computing have proven to be an essential tool for efficient and practical computation, and the story is not different for quantum computing. Indeed, there exists a large body of works that study advantages of parallel implementations of quantum gates for efficient quantum circuit implementations. Here, we focus on the recently invented efficient, arbitrary, simultaneously entangling (EASE) gates, available on a trapped-ion quantum computer. Leveraging its flexibility in selecting arbitrary pairs of qubits to be coupled with any degrees of entanglement, all in parallel, we show an n-qubit Clifford circuit can be implemented using 6log(n) EASE gates, an n-qubit multiply-controlled NOT gate can be implemented using 3n/2 EASE gates, and an n-qubit permutation can be implemented using six EASE gates. We discuss their implications to near-term quantum chemistry simulations and the state of the art pattern matching algorithm. Given Clifford + multiply-controlled NOT gates form a universal gate set for quantum computing, our results imply efficient quantum computation by EASE gates, in general.

The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 8% can be achieved.