The ability to simulate one Hamiltonian with another is an important primitive in quantum information processing. In this paper, a simulation method for arbitrary $σ_z \otimes σ_z$ interaction based on Hadamard matrices (quant-ph/9904100) is generalized for any pairwise interaction. We describe two applications of the generalized framework. First, we obtain a class of protocols for selecting an arbitrary interaction term in an n-qubit Hamiltonian. This class includes the scheme given in quant-ph/0106064v2. Second, we obtain a class of protocols for inverting an arbitrary, possibly unknown n-qubit Hamiltonian, generalizing the result in quant-ph/0106085v1.
We obtain the most general ensemble of qubits, for which it is possible to design a universal Hadamard gate. These states when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard `type' of operations and find ensembles of states for which such transformations hold. Unequal superposition of a qubit and its orthogonal complement is also investigated.
We propose a quantum state transfer from an atomic qubit to a cat-like qubit by means of one degenerate Raman interaction and one Hadamard gate operation for coherent states. We show that the coefficients of the atomic qubit can be mapped onto coherent state qubit, with an effective qubit state transfer.
We discuss and experimentally demonstrate a probabilistic Hadamard gate for coherent state qubits. The scheme is based on linear optical components, non-classical resources and the joint projective action of a photon counter and a homodyne detector. We experimentally characterize the gate for the coherent states of the computational basis by full tomographic reconstruction of the transformed output states. Based on the parameters of the experiment we simulate the fidelity for all coherent state qubits on the Bloch sphere.
We propose the implementation of a deterministic Hadamard gate for logical photonic qubits encoded in superpositions of coherent states of a harmonic oscillator. The proposed scheme builds on a recently introduced set of conditional operations in the strong dispersive regime of circuit QED [Z. Leghtas et al. Phys. Rev. A. {\bf 87} (2013)]. We further propose an architecture for coupling two such logical qubits and provide a universal set of deterministic quantum gates. Based on parameter values taken from the current state of the art, we give estimates for the achievable gate fidelities accoun…
We establish the non-existence of a universal Hadamard gate for arbitrary unknown qubits, by considering two different principles; namely, no-superluminal signalling and non-increase of entanglement under LOCC. It is also shown that these principles are not violated if and only if the qubit states are of a special form obtained in our recent work [ IJQI (in press), quant-ph/0505068 ].
We present an improved low-overhead implementation of surface code logical H. We describe in full detail logical H applied to a single distance 7 double-defect logical qubit in an otherwise idle scalable array such qubits. Our goal is to provide a clear description of logical H and to emphasize that the surface code possesses low-overhead implementations of the entire Clifford group.
We consider how the Hamiltonian Quantum Computing scheme introduced in [arXiv:1509.01278] can be implemented using a 2D array of superconducting transmon qubits. We show how the scheme requires the engineering of strong attractive cross-Kerr and weak flip-flop or hopping interactions and we detail how this can be achieved. Our proposal uses a new electric circuit for obtaining the attractive cross-Kerr coupling between transmons via a dipole-like element. We discuss and numerically analyze the forward motion and execution of the computation and its dependence on coupling strengths and their va…
In this paper, we develop a new classical simulation of quantum bit (qubit) by use of analog components in order to be able to simulate the quantum properties such as the superposition of states. As part of this new approach, we have also implemented many logical gates (Pauli-X, Hadamard, and Conditional phase-shift gates). Simulation of Hadamard and Conditional phase-shift gates using Proteus Design Suite is also presented and discussed in this work.
We demonstrate how insights gained from reformulating the problem of quantum teleportation into one of reversing quantum operations, and designing optimum completely positive maps for teleportation, can enable one to explore optimal approximate reversal of quantum operations on a single qubit. In particular, we show that the optimal approximate reversal of a generalized depolarizing channel can be achieved using only unitary transformations. We also show that for a quantum channel, which reveals some information about the input state, extremal completely positive maps and not unitary transform…
We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iSWAP gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verif…
By the standard second law of thermodynamics, heat spontaneously flows from a hotter body to a colder body. However, quantum systems in which quantum correlations play a prominent role can exhibit a non-classical reversal of such heat flow. We propose a quantum system consisting of a chain of qubits, each in local Gibbs states, where only adjacent qubits are allowed to thermally interact. By controlling initial quantum correlations along the chain, we then demonstrate non-classical heat reversal for the special case of a three-qubit chain on a quantum computer. We explore multiple initial cond…
We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-re…
We show that an n-th root of the Walsh-Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubi…
Controlled-NOT (CNOT) gates are commonly included in the standard gate set of quantum processors and provide an important way to entangle qubits. For fixed-frequency qubits using the cross-resonance entangling technique, using the higher-frequency qubit to control the lower-frequency qubit enables much shorter entangling times than using the lower-frequency qubit as the control. Consequently, when implementing a CNOT gate where logical control by the lower-frequency qubit is needed, compilers may implement this functionality by using an equivalent circuit such as placing Hadamard gates on both…
Spontaneous photon scattering by an atomic qubit is a notable example of environment-induced error and is a fundamental limit to the fidelity of quantum operations. In the scattering process the qubit loses its distinctive and coherent character owing to its entanglement with the photon. Using a single trapped ion we show that by utilizing the information carried by the photon we are able to coherently reverse this process and correct for the scattering error. We further used quantum process tomography to characterize the photon-scattering error and its correction scheme and demonstrate a corr…
The solving of linear systems provides a rich area to investigate the use of nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for skewed linear systems for over-determined and under-determined cases. Our input model is such that the columns or rows of the matrix defining the linear system are given via quantum circuits of poly-logarithmic depth and the number of circuits is much smaller than their Hilbert space dimension. Our algorithms have poly-logarithmic dependence on the dimension and polynomial dependence in other natu…
For n an even number of qubits and v a unitary evolution, a matrix decomposition v=k1 a k2 of the unitary group is explicitly computable and allows for study of the dynamics of the concurrence entanglement monotone. The side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. In this work, we provide an explicit numerical algorithm computing v=k1 a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument function, allowing for a theory of concurrence dynamics in odd qubits.…
Readout and retrieval processes are proposed for efficient, high-fidelity quantum state transfer between a matter qubit, encoded in the level structure of a single atom or ion, and a photonic qubit, encoded in a time-reversal-symmetric single-photon wave packet. They are based on controlling spontaneous photon emission and absorption of a matter qubit on demand in free space by stimulated Raman adiabatic passage. As these processes do not involve mode selection by high-finesse cavities or photon transport through optical fibers, they offer interesting perspectives as basic building blocks for …
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
As the number of qubits available on noisy quantum computers grows, it will become necessary to efficiently select a subset of physical qubits to use in a quantum computation. For any given quantum program and device there are many ways to assign physical qubits for execution of the program, and assignments will differ in performance due to the variability in quality across qubits and entangling operations on a single device. Evaluating the performance of each assignment using fidelity estimation introduces significant experimental overhead and will be infeasible for many applications, while r…
Time-Reversal Selection Rules for Quantum Error Correction
Quantum Initiative, Florida State University（佛罗里达州立大学量子倡议）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Department of Mathematics, University of California, San Diego（加州大学圣地亚哥分校数学系）
The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.
We demonstrate the conditional reversal of a weak (partial-collapse) quantum measurement on a photonic qubit. The weak quantum measurement causes a nonunitary transformation of a qubit which is subsequently reversed to the original state after a successful reversing operation. Both the weak measurement and the reversal operation are implemented linear optically. The state recovery fidelity, determined by quantum process tomography, is shown to be over 94% for partial-collapse strength up to 0.9. We also experimentally study information gain due to the weak measurement and discuss the role of t…
We introduce a trichromatic graphical calculus for quantum computing. The generators represent three complementary observables that are treated on equal footing, hence reflecting the symmetries of the Bloch sphere. We derive the Euler angle decomposition of the Hadamard gate within it as well as the so-called supplementary relationships, which are valid equations for qubits that were not derivable within Z/X-calculus of Coecke and Duncan. More specifically, we have: dichromatic Z/X-calculus + Euler angle decomposition of the Hadamard gate = trichromatic calculus.
Time reversal of complex evolution on a quantum computer
Mindbreeze GmbH（Mindbreeze有限公司）；Basic Research Community for Physics e.V.（基础物理研究社区协会）；Univ Toulouse, CNRS, Laboratoire de Physique Théorique（图卢兹大学、法国国家科学研究中心、理论物理实验室）
In this paper, different from the other existing methods that only the special and limited numbers of qubits transmit between Alice and Bob, a general method is proposed to implement Bidirectional Quantum Controlled/uncontrolled Teleportation (BQCT/BQT) that Alice and Bob can transmit each type of arbitrary n and m-qubits to each other, simultaneously. This protocol is based on Controlled-NOT (CNOT) operation, appropriate single-qubit unitary operations and single-qubit measurement in the Z-basis and X-basis. Also, the quantum channel can be created by using Hadamard and CNOT operations. One o…
Recent investigations show that conservation laws limit the accuracy of gate operations in quantum computing. The inevitable error under the angular momentum conservation law has been evaluated so far for the CNOT, Hadamard, and NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we extend the above results to general single-qubit gates. We obtain an upper bound of the gate fidelity of arbitrary single-qubit gates implemented under arbitrary conservation laws, determined by the geometry of the conservation law and the gate operation on the Bloch sphere as well as the si…
We expand the time reversal symmetry arguments of quantum mechanics, originally proposed by Wigner in the context of unitary dynamics, to contain situations including generalized measurements for monitored quantum systems. We propose a scheme to derive the time reversed measurement operators by considering the Schrödinger picture dynamics of a qubit coupled to a measuring device, and show that the time reversed measurement operators form a Positive Operator Valued Measure (POVM) set. We present three particular examples to illustrate time reversal of measurement operators: (1) the Gaussian spi…
In this paper, we propose a scheme to enhance trapping of entanglement of two qubits in the environment of a photonic band gap material. Our entanglement trapping promotion scheme makes use of combined weak measurements and quantum measurement reversals. The optimal promotion of entanglement trapping can be acquired with a reasonable finite success probability by adjusting measurement strengths.
A general two qubit system expressed in terms of the complete set of unit and fifteen traceless, Hermitian Dirac matrices, is shown to exhibit novel features of this system. The well-known physical interpretations associated with the relativistic Dirac equation involving the symmetry operations of time-reversal T, charge conjugation C, parity P, and their products are reinterpreted here by examining their action on the basic Bell states. The transformation properties of the Bell basis states under these symmetry operations also reveal that C is the only operator that does not mix the Bell stat…
A protocol of quantum dense coding protection of two qubits is proposed in amplitude damping (AD) channel using weak measurement and reversal measurement. It is found that the capacity of quantum dense coding under the weak measurement and reversal measurement is always greater than that without weak measurement and reversal measurement. When the protocol is applied, for the AD channels with different damping coefficient, the result reflects that quantum entanglement can be protected and quantum dense coding becomes successful.
The uncertainty principle is an inherent characteristic of quantum mechanics. This principle can be formulated in various form. Fundamentally, this principle can be expressed in terms of the standard deviation of the measured observables. In quantum information theory the preferred mathematical quantity to express the entropic uncertainty relation is the Shannon's entropy. In this work, we consider the generalized entropic uncertainty relation in which there is an additional particle as a quantum memory. Alice measures on her particle $A$ and Bob, with memory particle $B$, predicts the Alice's…
We establish a novel procedure to analyze the entanglement properties of extremal density matrices depending on the parameters of a finite dimensional Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could exhibit Kramers degeneracy. This is done through the extremal density matrix formalism, which allows to extend the conventional variational principle to mixed states. By applying the positive partial transpose criterion in terms of the Correlation and Schlienz-Mahler matrices on the extremal density matrices, we demonstrate that it is possible to reach both pure and mixed e…
Nonstoquastic Hamiltonians are hard to simulate due to the sign problem in quantum Monte Carlo simulation. It is however unclear whether nonstoquasticity can lead to advantage in quantum annealing. Here we show that YY-interaction between the qubits makes the adiabatic path during quantum annealing, and therefore the performance, dependent on spin-reversal transformations. With the right choice of spin-reversal transformation, a nonstoquastic Hamiltonian with YY-interaction can outperform stoquastic Hamiltonians with similar parameters. We introduce an optimization protocol to determine the op…
We demonstrate the ability to control the spontaneous emission from a superconducting qubit coupled to a cavity. The time domain profile of the emitted photon is shaped into a symmetric truncated exponential. The experiment is enabled by a qubit coupled to a cavity, with a coupling strength that can be tuned in tens of nanoseconds while maintaining a constant dressed state emission frequency. Symmetrization of the photonic wave packet will enable use of photons as flying qubits for transfering the quantum state between atoms in distant cavities.
In the presence of time-reversal symmetry, quantum interference gives strong corrections to the electric conductivity of disordered systems. The self-interference of an electron wavefunction traveling time-reversed paths leads to effects such as weak localization and universal conductance fluctuations. Here, we investigate the effects of broken time-reversal symmetry in a driven artificial two-level system. Using a superconducting flux qubit, we implement scattering events as multiple Landau-Zener transitions by driving the qubit periodically back and forth through an avoided crossing. Interfe…
