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 able to accomplish them. This classifies two--qubit gates into four categories, differing in their capability to transmit classical, as well as quantum, bits of information.
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 l…
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.
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…
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.
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.
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…
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…
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.
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…
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…
Ying conceived of using two or more small-capacity quantum computers to produce a larger-capacity quantum computing system by quantum parallel programming ([M. S. Ying, Morgan-Kaufmann, 2016]). In doing so, the main obstacle is separating the quantum gates in the whole circuit to produce a tensor product of the local gates. It has been showed that there are few separable multipartite quantum gates, so the approximate separation problem involves finding local quantum gates that approximate a given inseparable gate. We propose and study a problem involving the approximate separation of multipart…
Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological …
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local-invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root …
We present a method of optimizing recently designed protocols for implementing an arbitrary nonlocal unitary gate acting on a bipartite system. These protocols use only local operations and classical communication with the assistance of entanglement, and are deterministic while also being "one-shot", in that they use only one copy of an entangled resource state. The optimization is in the sense of minimizing the amount of entanglement used, and it is often the case that less entanglement is needed than with an alternative protocol using two-way teleportation.
We investigate the feasibility of single-shot Toffoli- and Fredkin-gate realizations in qubit arrays with Heisenberg-type exchange interactions between adjacent qubits. As follows from the Lie-algebraic criteria of controllability, such an array is rendered completely controllable -- equivalent to allowing universal quantum computation -- by a Zeeman-like control field with two orthogonal components acting on a single "actuator" qubit. Adopting this local-control setting, we start our analysis with piecewise-constant control fields and determine the global maxima of the relevant figure of meri…
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 experimentally demonstrate, with an all-optical setup, a nonlocal deterministic quantum non-demolition interaction gate applicable to quantum states at nodes separated by a physical distance and connected by classical communication channels. The gate implementation, based on entangled states shared in advance, local operations, and classical communication, runs completely in parallel fashion at both the local nodes, requiring minimum resource. The nondemolition character of the gate up to the local unitary squeezing is verified by the analysis using several coherent states. A genuine quantu…
In the circuit model, quantum computers rely on the availability of a universal quantum gate set. A particularly intriguing example is a set of two-qubit only gates: matchgates, along with SWAP (the exchange of two qubits). In this paper, we show a simple decomposition of arbitrary matchgates into better known elementary gates, and implement a matchgate in a linear-optics experiment using single photons. The gate performance was fully characterized via quantum process tomography. Moreover, we represent the resulting reconstructed quantum process in a novel way, as a fidelity map in the space o…
It is well known that local gates have smaller error than non-local gates. For this reason it is natural to make two states equivalent if they differ by a local gate. Since two states that differ by a local gate have the same entanglement entropy, then the entanglement entropy defines a function in the quotient space. In this paper we study this equivalence relation on (i) the set $\mathbb{R}\hbox{Q}(3)$ of 3-qubit states with real amplitudes, (ii) the set $Q\mathcal{C}$ of 3-qubit states that can be prepared with gates on the Clifford group, and (iii) the set $Q\mathbb{R}\mathcal{C}$ of 3-qub…
We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking gene…
Trapped-ion quantum computers have demonstrated high-performance gate operations in registers of about ten qubits. However, scaling up and parallelizing quantum computations with long one-dimensional (1D) ion strings is an outstanding challenge due to the global nature of the motional modes of the ions which mediate qubit-qubit couplings. Here, we devise methods to implement scalable and parallel entangling gates by using engineered localized phonon modes. We propose to tailor such localized modes by tuning the local potential of individual ions with programmable optical tweezers. Localized mo…
We demonstrate an optical gate that increases the size of polarization-based W states by accessing only one of the qubits. Using this gate, we have generated three-photon and four-photon W states with fidelities $0.836\pm 0.042 $ and $0.784\pm 0.028$, respectively. We also confirmed existence of pairwise entanglement in every pair of the qubits including the one that was left untouched by the gate. The gate is applicable to any size of W states and hence is a universal tool for expanding entanglement.
This note proves that arbitrary local gates together with any entangling bipartite gate V are universal. Previously this was known only when access to both V and V^{-1} was given, or when approximate universality was demanded.
Dynamics of many-qubit systems, that may correspond to computational processing with quantum systems, can be efficiently and generally approximated by a sequence of two- and single-qubit gates. In practical applications, however, a quantum gate prepared as a unitary transformation may appear as a noisy channel and consequently may inhibit quantum advantages. In this work, we apply the scheme of channel discrimination to detect if a quantum gate that is actually realized is unitary or noisy. We show that a two-qubit unitary transformation and its noisy counterpart can be optimally discriminated…
In this paper, we demonstrate n-party controlled unitary gate implementations locally on arbitrary remote state through linear entangled channel where control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party. In such a network, we describe the protocol of simultaneous implementation of controlled-Hermitian gate starting from three party scenario. We also explicate the implementation of three party controlled-Unitary gate, a generalized form of Toffoli gate and subsequently generalize the protocol for n-party using minim…
A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon los…
We propose a simple probabilistic optical gate to expand polarization entangled W states. The gate uses one polarization-dependent beamsplitter and a horizontally polarized single photon as an ancilla. The gate post-selectively expands $N$-photon W states to $(N+1)$-photon W states. A feasibility analysis considering the realistic experimental conditions show that the scheme is within the reach of the current quantum optical technologies.
Evolution-Level Quantum Optimal Control of Single-Qubit Gates with Physics-Informed Neural Networks
School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理与信息工程学院）；School of Science, Xi’an University of Posts and Telecommunications（西安邮电大学理学院）
