In this paper we study the properties of two-qubit gates. We review the most common parameterizations for the local equivalence classes of two-qubit gates and the connections between them. We then introduce a new discrete local invariant, namely the number of local degrees of freedom that a gate can bind. The value of this invariant is calculated analytically for all the local equivalence classes of two-qubit gates. We find that almost all two-qubit gates can bind the full six local degrees of freedom and are in this sense more effective than the controlled-NOT gate, which only can bind four local degrees of freedom.
We show a simple relation connecting entangling power and local invariants of two-qubit gates. From the relation, a general condition under which gates have same entangling power is arrived. The relation also helps in finding the lower bound of entangling power for perfect entanglers, from which the classification of gates as perfect and nonperfect entanglers is obtained in terms of local invariants.
A gauge-invariant theory of small Markovian errors in quantum gate sets
Quantum Performance Laboratory, Sandia National Laboratories（桑迪亚国家实验室量子性能实验室）；Department of Physics and Astronomy, Quantum New Mexico Institute, University of New Mexico（新墨西哥大学量子新墨西哥学院物理与天文学系）；IonQ
We propose a new strategy to physically implement a universal set of quantum gates based on geometric phases accumulated in the nondegenerate eigenstates of a designated invariant operator in a periodic physical system. The system is driven to evolve in such a way that the dynamical phase shifts of the invariant operator eigenstates are the same (or {\it mod} $2π$) while the corresponding geometric phases are nontrivial. We illustrate how this strategy to work in a simple but typical NMR-type qubit system.
Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which are necessary for fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, weakening the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolution via invariant-based shortcuts, where universal quantum gates can be induced in one step without path mutation and the gate time is also effectively shortened. …
Among existing approaches to holonomic quantum computing, the adiabatic holonomic quantum gates (HQGs) suffer errors due to decoherence, while the non-adiabatic HQGs either require additional Hilbert spaces or are difficult to scale. Here, we report a systematic, scalable approach based on dynamical invariants to realize HQGs without using additional Hilbert spaces. While presenting the theoretical framework of our approach, we design and experimentally evaluate single-qubit and two-qubits HQGs for the nuclear magnetic resonance system. The single-qubit gates acquire average fidelity 0.9972 by…
We analyse the implementation of a fast nonadiabatic CZ gate between two transmon qubits with tuneable coupling. The gate control method is based on a theory of dynamical invariants which leads to reduced leakage and robustness against decoherence. The gate is based on a description of the resonance between the $|11\rangle$ and $|20\rangle$ using an effective Hamiltonian with the 6 lowest energy states. A modification of the invariants method allows us to take into account the higher-order perturbative corrections of this effective model. This enables a gate fidelity several orders of magnitud…
We extend recent work on a leakage-protected, adiabatic entangling gate for exchange-only spin qubits [Doherty and Wardrop, PRL 111, 050503 (2013)] by adapting to a setting where single spins are not assumed to be polarized on preparation. Previous gate constructions do not function correctly when "gauge spins" are uninitialized, because the entangling gate has different, non-trivial action in different gauge subspaces. Our construction inherits many of the desirable features of the previous work while addressing the gauge-dependent behavior. Using numerical simulation, we show that the result…
Entanglement of two parts of a quantum system is a non-local property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of non-local properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-bit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can …
We theoretically propose an alternative method to realize a deterministic SWAP gate using shortcuts to adiabatic passage based on the approach of Lewis- Riesenfeld invariants in cavity quantum electronic dynamics (QED). By combining Lewis-Riesenfeld invariants with quantum Zeno dynamics, the SWAP gate can be achieved deterministically. The strict numerical results show that our scheme is a fast and robust approach to achieve SWAP gate.
We investigate the groups generated by the sets of $CP$, $CNOT$ and $SWAP^α$ (power-of-SWAP) quantum gate operations acting on $n$ qubits. Isomorphisms to standard groups are found, and using techniques from representation theory, we are able to determine the invariant subspaces of the $n-$qubit Hilbert space under the action of each group. For the $CP$ operation, we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order $2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to the computational state-vectors. For the $CNOT$ operation, we find isomorphi…
We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.
We study the minimum time to implement an arbitrary two-qubit gate in two heteronuclear spins systems. We give a systematic characterization of two-qubit gates based on the invariants of local equivalence. The quantum gates are classified into four classes, and for each class the analytical formula of the minimum time to implement the quantum gates is explicitly presented. For given quantum gates, by calculating the corresponding invariants one easily obtains the classes to which the quantum gates belong. In particular, we analyze the effect of global phases on the minimum time to implement th…
Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.
We derive an integral expression for the filter-transfer function of an arbitrary one-qubit gate through the use of dynamical invariant theory and Hamiltonian reverse engineering. We use this result to define a cost function which can be efficiently optimized to produce one-qubit control pulses that are robust against specified frequency bands of the noise power spectral density. We demonstrate the utility of our result by generating optimal control pulses that are designed to suppress broadband detuning and pulse amplitude noise. We report an order of magnitude improvement in gate fidelity in…
Spectator Leakage Suppression via Invariant Subspace Engineering for CZ Gates in Superconducting Quantum Circuits
University of Science and Technology of China（中国科学技术大学）；CAS Center For Excellence in Quantum Information and Quantum Physics（中国科学院量子信息与量子物理前沿卓越创新中心）；Suzhou Institute for Advanced Research, University of Science and Technology of China（中国科学技术大学苏州高等研究院）；Institute of Artificial Intelligence, Hefei Comprehensive National Science Center（合肥综合性国家科学中心人工智能研究院）
We characterize the quantum gate fidelity in a state-independent manner by giving an explicit expression for its variance. The method we provide can be extended to calculate all higher order moments of the gate fidelity. Using these results we obtain a simple expression for the variance of a single qubit system and deduce the asymptotic behavior for large-dimensional quantum systems. Applications of these results to quantum chaos and randomized benchmarking are discussed.
This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the presence of local unitary transformations. We show that this same R generates a new non-trivial invariant of braids, knots, and links. Other solutions of the Yang-Baxter Equation are also shown to be universal for quantum computation. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss qu…
We propose an alternative scheme of shortcuts to quantum phase gate in a much shorter time based on the approach of Lewis-Riesenfeld invariants in cavity quantum electronic dynamics (QED) systems. This scheme can be used to perform one-qubit phase gate, two-qubit controlled phase gate and also multiqubit controlled phase gate. The strict numerical simulation for some quantum gates are given, and demonstrate that the total operation time of our scheme is shorter than previous schemes and very robustness against decoherence.
We study the computation power of lattices composed of two dimensional systems (qubits) on which translationally invariant global two-qubit gates can be performed. We show that if a specific set of 6 global two qubit gates can be performed, and if the initial state of the lattice can be suitably chosen, then a quantum computer can be efficiently simulated
We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.
Adiabatic quantum gate implementation generally takes longer time, which is disadvantageous in view of decoherence. In this report we implement several essential one-qubit quantum gates nonadiabatically by making use of a dynamical invariant associated with a Hamiltonian. Moreover we require that these gates be holonomic, that is, the dynamical phases associated with the gates vanish. Our implementation is based on our recent work [J. Phys. Soc. Jpn. 83, 034001 (2014)] and the gate parameters required for the implementations are found by numerical optimization.
We investigate the minimal resources that are required in the local implementation of non-local quantum gates in a distributed quantum computer. Both classical communication requirements and entanglement consumption are investigated. We present general statements on the minimal resource requirements and present optimal procedures for a number of important gates, including CNOT and Toffoli gates. We show that one bit of classical communication in each direction is both necessary and sufficient for the non-local implementation of the quantum CNOT, while in general two bits in each direction is r…
Geometric Quantum Gates of Non-closed Paths Under Counterdiabatic Driving
College of Physics and Electronic Engineering, Shanxi University（山西大学物理电子工程学院）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）
Two different measures of operator entanglement of two-qubit gates, namely, Schmidt strength and linear entropy, are studied. While these measures are shown to have one-to-one relation between them for Schmidt number 2 class of gates, no such relation exists for Schmidt number 4 class, implying that the measures are inequivalent in general. Further, we establish a simple relation between linear entropy and local invariants of two-qubit gates. The implication of the relation is discussed.
We implement faster-than-adiabatic two-qubit phase gates using smooth state-dependent forces. The forces are designed to leave no final motional excitation, independently of the initial motional state in the harmonic, small-oscillations limit. They are simple, explicit functions of time and the desired logical phase of the gate, and are based on quadratic invariants of motion and Lewis-Riesenfeld phases of the normal modes.
We introduce the interaction cost of a non-local gate as the minimal time of interaction required to perform the gate when assisting the process with fast local unitaries. This cost, of interest both in the areas of quantum control and quantum information, depends on the specific interaction, and allows to compare in an operationally meaningful manner any two non-local gates. In the case of a two-qubit system, an analytical expression for the interaction cost of any unitary operation given any coupling Hamiltonian is obtained. One gate may be more time-consuming than another for any possible i…
On the local equivalence of trapped-ion two-qudit gates
P.N. Lebedev Physical Institute of the Russian Academy of Sciences（俄罗斯科学院列别捷夫物理研究所）；Russian Quantum Center（俄罗斯量子中心）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
Local implementation of non-local quantum gates is necessary in a distributed quantum computer. Here, we demonstrate the non-local implementation of controlled-unitary quantum gates proposed by Eisert et al. (Phys Rev A 62:052317, 2000) using the five-qubit IBM quantum computer. We verify the fidelity and accuracy of the implementation through the techniques of quantum state and process tomographies.
The problems related to the management of large quantum registers could be handled in the context of distributed quantum computation: unitary non-local transformations among spatially separated local processors are realized performing local unitary transformations and exchanging classical communication. In this paper, we propose a scheme for the implementation of universal non-local quantum gates such as a controlled-$\gate{NOT}$ ($\cnot$) and a controlled-quantum phase gate ($\gate{CQPG}$). The system we have chosen for their physical implementation is a Cavity-Quantum-Electro-Dynamics (CQED)…
A general quantum network for implementing non-local control-unitary gates, between remote parties at minimal entanglement cost, is shown to be a rooted-tree structure. Starting from a five party scenario, we demonstrate the local implementation of simultaneous control-Hermitian and multiparty control-unitary gates in an arbitrary n-party network. Previously established networks are shown to be special cases of this general construct.
Many quantum algorithms can be written as a composition of unitaries, some of which can be exactly synthesized by a universal fault-tolerant gate set, while others can be approximately synthesized. A quantum compiler synthesizes each approximately synthesizable unitary up to some approximation error, such that the error of the overall unitary remains bounded by a certain amount. In this paper we consider the case when the errors are measured in the global phase invariant distance. Apart from deriving a relation between this distance and the Frobenius norm, we show that this distance composes. …
The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.
Nonlocal two-qubit gates are geometrically represented by tetrahedron known as Weyl chamber within which perfect entanglers form a polyhedron. We identify that all edges of the Weyl chamber and polyhedron are formed by single parametric gates. Nonlocal attributes of these edges are characterized using entangling power and local invariants. In particular, SWAP (power)alpha family of gates constitutes one edge of the Weyl chamber with SWAP-1/2 being the only perfect entangler. Finally, optimal constructions of controlled-NOT using SWAP-1/2 gate and gates belong to three edges of the polyhedron a…
A non-local unitary transformation of two qubits occurs when some Hamiltonian interaction couples them. Here we characterize the amount, as measured by time, of interaction required to perform two--qubit gates, when also arbitrarily fast, local unitary transformations can be applied on each qubit. The minimal required time of interaction, or interaction cost, defines an operational notion of the degree of non--locality of gates. We characterize a partial order structure based on this notion. We also investigate the interaction cost of several communication tasks, and determine which gates are …
Optimal implementation of quantum gates is crucial for designing a quantum computer. The necessary condition for optimal construction of a two-qubit unitary operation is obtained. It can be proved that the B gate is the unique gate that can construct a two-qubit universal circuit with only two applications, i.e. this condition is also sufficient in the case of two applications of the elementary two-qubit gate. It is also shown that one half of perfect entanglers can not simulate an arbitrary two-qubit gate with only 3 applications.
Motivated by experimental limitations commonly met in the design of solid state quantum computers, we study the problems of non-local Hamiltonian simulation and non-local gate synthesis when only homogeneous local unitaries are performed in order to tailor the available interaction. Homogeneous (i.e. identical for all subsystems) local manipulation implies a more refined classification of interaction Hamiltonians than the inhomogeneous case, as well as the loss of universality in Hamiltonian simulation. For the case of symmetric two-qubit interactions, we provide time-optimal protocols for bot…
We propose a scalable design of entangling quantum gates for large ion crystals with the following desirable features: 1) The gate design is universal and applicable for large ion crystals of arbitrary sizes; 2) The gate has no speed limitation and can work outside of the Lamb-Dicke region; 3) The gate operates by driving from either continuous-wave or pulsed laser beams; 4) The gate is insensitive to slow variation of the laser optical phase and works under a thermal state for the ions' motion; 5) The intrinsic gate infidelity can be reduced to a level well below the threshold for fault-toler…
For a generic $n$-qubit system, local invariants under the action of $SL(2,\mathbb{C})^{\otimes n}$ characterize non-local properties of entanglement. In general, such properties are not immediately apparent and hard to construct. Here we consider certain two-qubit Yang-Baxter operators, which we dub of the `X-type', and show that their eigenvalues completely determine the non-local properties of the system. Moreover, we apply the Turaev procedure to these operators and obtain their associated link/knot polynomials. We also compute their entangling power and compare it with that of a generic t…
I discuss how to perform fault-tolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when next-to-nearest-neighbor gates are available, and present explicit constructions. In two or three dimensions, I also show how nearest-neighbor gates can give a threshold result. In all cases, I simply demonstrate that a threshold exists, and do not attempt to optimize the error correction circuit or determine the exact value of the threshold. The additional overhead due to the faul…
Verifying the correct functioning of quantum gates is a crucial step towards reliable quantum information processing, but it becomes an overwhelming challenge as the system size grows due to the dimensionality curse. Recent theoretical breakthroughs show that it is possible to verify various important quantum gates with the optimal sample complexity of $O(1/ε)$ using local operations only, where $ε$ is the estimation precision. In this work, we propose a variant of quantum gate verification (QGV) which is robust to practical gate imperfections, and experimentally realize efficient QGV on a two…
A scheme of universal quantum computation on a chain of qubits is described that does not require local control. All the required operations, an Ising-type interaction and spatially uniform simultaneous one-qubit gates, are translation-invariant.
We propose a scheme to implement high-fidelity conditional phase gates on pair of trapped ions immersed in a two-dimensional Coulomb crystal, using interaction mediated by all axial modes without side-band addressing. We show through numerical calculations that only local modes can be excited to achieve entangling gates through shaping the laser beams, so that the complexity of the quantum gate does not increase with the size of the system. These results suggest a promising approach for realization of large scale fault-tolerant quantum computation in two dimensional traps architecture.
We say that collection of $n$-qudit gates is universal if there exists $N_0\geq n$ such that for every $N\geq N_0$ every $N$-qudit unitary operation can be approximated with arbitrary precision by a circuit built from gates of the collection. Our main result is an upper bound on the smallest $N_0$ with the above property. The bound is roughly $d^8 n$, where $d$ is the number of levels of the base system (the '$d$' in the term qu$d$it.) The proof is based on a recent result on invariants of (finite) linear groups.
With the development of controllable quantum systems, fast and practical characterization for multi-qubit gates is essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking asks for the complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. In this work, we propose a character-cycle benchmarking protocol and a character-average benchmarking protocol only using local twirling gates to estimate the process fide…
Non-local higher-energy auxiliary states have been successfully used to entangle pairs of qubits in different quantum computing systems. Typically a longer-span non-local state or sequential application of few-qubit entangling gates are needed to produce a non-trivial multiqubit gate. In many cases a single non-local state that span over the entire system is difficult to use due to spectral crowding or impossible to have. At the same time, many multiqubit systems can naturally develop a network of multiple non-local higher-energy states that span over few qubits each. We show that continuous t…
The simulation of time evolution of large quantum systems is a classically challenging and in general intractable task, making it a promising application for quantum computation. A Trotter-Suzuki approximation yields an implementation thereof, where a higher approximation accuracy can be traded for an increased gate count. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of translationally invariant quantum systems. Our strategy can improve upon the Trotter-Suzuki accuracy by several orde…
We study non-local two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of non-local gates is a 3-Torus. We derive the invariants for local transformations, and connect these local invariants to the coordinates of the 3-Torus. Since different points on the 3-Torus may correspond to the same local equivalence class, we use the Weyl group theory to reduce the symmetry. We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. We…
The formation of multipartite quantum entanglement by repeated operation of one and two qubit gates is examined. The resulting entanglement is evaluated using two measures: the average bipartite entanglement and the Groverian measure. A comparison is made between two geometries of the quantum register: a one dimensional chain in which two-qubit gates apply only locally between nearest neighbors and a non-local geometry in which such gates may apply between any pair of qubits. More specifically, we use a combination of random single qubit rotations and a fixed two-qubit gate such as the control…
Efficient verification of the functioning of quantum devices is a key to the development of quantum technologies, but is a daunting task as the system size increases. Here we propose a simple and general framework for verifying unitary transformations that can be applied to both individual quantum gates and gate sets, including quantum circuits. This framework enables efficient verification of many important unitary transformations, including but not limited to all bipartite unitaries, Clifford unitaries, generalized controlled-$Z$ gates, generalized controlled-NOT gates, the controlled-SWAP g…
