Yuji Tanaka, Tsubasa Ichikawa, Masahito Tada-Umezaki, Yukihiro Ota, Mikio Nakahara
We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.
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Linear-depth quantum oracles for clique problems from edge colorings and graph states, with linear non-Clifford cost and a provably bounded-error k-clique search
本文利用完全图的1-因子分解设计深度为O(n)的量子预言机，将三角形查找复杂度降至O(n^2.25 poly(log n))，并基准测试了求解k-CLIQUE所需的振幅放大迭代次数。
We propose a new quantum circuit for the quantum search problem. The quantum circuit is superior to Grover's algorithm in some realistic cases. The reasons for the superiority are in short as follows: In the quantum circuit proposed in this paper, all the operators except for the oracle can be written as direct products of single-qubit gates. Such separable operators can be executed much faster than multi-particle operators, such as c-NOT gates and Toffoli gates, in many realistic systems. The idea of this quantum circuit is inspired by the Hamiltonian used in the adiabatic quantum computer. I…
We present a constructive method to create quantum circuits that implement oracles $|x\rangle|y\rangle|0\rangle^k \mapsto |x\rangle|y \oplus f(x)\rangle|0\rangle^k$ for $n$-variable Boolean functions $f$ with low $T$-count. In our method $f$ is given as a 2-regular Boolean logic network over the gate basis $\{\land, \oplus, 1\}$. Our construction leads to circuits with a $T$-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for $T$ gates, and explore the space/complexity trade-off of quantum circui…
p-SWAP: A Generic Cost-Effective Quantum Boolean-Phase SWAP Gate Using Two CNOT Gates and the Bloch Sphere Approach
本研究提出通用低成本的p-SWAP量子布尔相位交换门，仅用2个CNOT门，通过布洛赫球方法几何选取可定制相位p，相较标准SWAP门实现约23%量子成本降低与26%深度缩减。
An investigation of an optimal universal unitary Controlled-NOT gate that performs a specific operation on two unknown states of qubits taken from a great circle of the Bloch sphere is presented. The deep analogy between the optimal universal C-NOT gate and the `equatorial' quantum cloning machine (QCM) is shown. In addition, possible applications of the universal C-NOT gate are briefly discussed.
The universal quantum cloning machine and the universal NOT gate acting on a single qubit can be implemented very generally by slightly modifying the protocol of quantum state teleportation. The experimental demonstration of the 1 to 2 cloning process according to the above scheme has been realized for a qubit encoded in photon polarization.
We construct a Universal Quantum Entanglement Concentration Gate (QEC-Gate). Special times operations of QEC-Gate can transform a pure 2-level bipartite entangled state to nearly maximum entanglement. The transformation can attain any required fidelity with optimal probability by adjusting concentration step. We also generate QEC-Gate to the Schmidt decomposable multi-partite system.
Constructing a set of universal quantum gates is a fundamental task for quantum computation. The existence of noises, disturbances and fluctuations is unavoidable during the process of implementing quantum gates for most practical quantum systems. This paper employs a sampling-based learning method to find robust control pulses for generating a set of universal quantum gates. Numerical results show that the learned robust control fields are insensitive to disturbances, uncertainties and fluctuations during the process of realizing universal quantum gates.
Quantum oracles play key roles in the studies of quantum computation and quantum information. But implementing quantum oracles efficiently with universal quantum gates is a hard work. Motivated by genetic programming, this paper proposes a novel approach to evolve quantum oracles with a hybrid quantum-inspired evolutionary algorithm. The approach codes quantum circuits with numerical values and combines the cost and correctness of quantum circuits into the fitness function. To speed up the calculation of matrix multiplication in the evaluation of individuals, a fast algorithm of matrix multipl…
We prove the existence of a class of two--input, two--output gates any one of which is universal for quantum computation. This is done by explicitly constructing the three--bit gate introduced by Deutsch [Proc.~R.~Soc.~London.~A {\bf 425}, 73 (1989)] as a network consisting of replicas of a single two--bit gate.
A universal quantum gate is introduced for tensors of vector spaces. By using integer powers of such a gate and by using classical reversible gates one can approximate any element of the unitary group to any accuracy needed. The proof uses a version of Kronecker's theory and the structure of the Bloch sphere for tensors.
By a significant modification of the standard protocol of quantum state Teleportation two processes ''forbidden'' by quantum mechanics in their exact form, the Universal NOT gate and the Universal Optimal Quantum Cloning Machine, have been implemented contextually and optimally by a fully linear method. In particular, the first experimental demonstration of the Tele-UNOT Gate, a novel quantum information protocol has been reported (cfr. quant-ph/0304070). A complete experimental realization of the protocol is presented here.
To generate arbitrary one- and two-qubit gates, the universal decompositions are usually used in quantum computing, and the universality of these decompositions has been demonstrated. However, in realistic experiments, gate errors may affect the universality of the universal decompositions. Here, we focus on the single-qubit-gate decomposition scheme and study the coherent-error effects on universality. We prove that, in the parameter space which we studied, some kinds of coherent errors will not affect the original universality, but some others will destroy it. We provide the definition and a…
How to implement quantum oracle with limited resources raises concerns these days. We design two ancilla-adjustable and efficient algorithms to synthesize SAT-oracle, the key component in solving SAT problems. The previous work takes 2m-1 ancillary qubits and O(m) elementary gates to synthesize an m clauses oracle. The first algorithm reduces the number of ancillary qubits to 2\sqrt{m}, with at most an eightfold increase in circuit size. The number of ancillary qubits can be further reduced to 3 with a quadratic increase in circuit size. The second algorithm aims to reduce the circuit depth. B…
We present a universal set of quantum gate operations based on exchange-only spin qubits in a double quantum dot, where each qubit is obtained by three electrons in the (2,1) filling. Gate operations are addressed by modulating electrostatically the tunneling barrier and the energy offset between the two dots, singly and doubly occupied respectively. We propose explicit gate sequences of single qubit operations for Hadamard gate and $π$/8 gate, and the two-qubit controlled NOT (CNOT) gate, to complete the universal set. The unswitchable interaction between the two electrons of the doubly occup…
The controlled-NOT gate and controlled square-root NOT gate play an important role in quantum algorithm. This article reports the experimental results of these two universal quantum logic gates (controlled square-root NOT gate and controlled-NOT gate) on a 7-qubit NMR quantum computer. Further, we propose a simple experimental method to measure and correct the error in the controlled phase-shift gate, which is helpful to construct a more perfect phase-shift gate experimentally and can also be used in more qubits discrete Fourier transformation.
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controll…
Benchmarking of universal qutrit gates
针对qutrit系统研究需求，该研究提出基于超二面体群的通用qutrit门集合表征方案，其T门基准测试资源需求与现有Clifford随机基准测试相近，结合后可实现通用qutrit门集的完整表征。
Twisted rapid passage is a type of non-adiabatic rapid passage that gives rise to controllable quantum interference effects that were first observed experimentally in 2003. We show that twisted rapid passage sweeps can be used to implement a universal set of quantum gates that operate with high-fidelity. For each gate in the universal set, sweep parameter values are provided which simulations indicate will yield a quantum gate with error probability P_{e} < 10**(-4). Note that all gates in this universal set are driven by a **single** type of control field (twisted rapid passage), and the erro…
Universal quantum gates lie at the heart of designing quantum computer. We construct two compact quantum circuits to implement post-selected controlled-phase-flip (CPF) gate and Toffoli gate with linear optics assisted by one and two single photons, respectively. The current existing maximum success probability of 1/4 for linear optical CPF gate is achieved by resorting to an ancillary single photon rather than an entangled photon pair or two single photons. Remarkably, our Toffoli gate is accomplished with current maximum success probability of 1/30 without using additional entangled photon p…
Twisted rapid passage is a type of non-adiabatic rapid passage that generates controllable quantum interference effects that were first observed experimentally in 2003. It is shown that twisted rapid passage sweeps can be used to implement a universal set of quantum gates that operate with high-fidelity. The gate set consists of the Hadamard and NOT gates, together with variants of the phase, pi/8, and controlled-phase gates. For each gate g in the universal set, sweep parameter values are provided which numerical simulations indicate will produce a unitary operation that approximates g with e…
Coherent Rollout Oracles for Finite-Horizon Sequential Decision Problems
本文提出一种相干展开预言，通过可逆电路分析，实现有限时间序贯决策问题的高效解决，结合可逆转换和谓词评估电路，给出多项式规模的相干展开预言，并验证其在最佳臂流水线中的应用。
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.
We show, within the circuit model, how any quantum computation can be efficiently performed using states with only real amplitudes (a result known within the Quantum Turing Machine model). This allows us to identify a 2-qubit (in fact 2-rebit) gate which is universal for quantum computing, although it cannot be used to perform arbitrary unitary transformations.
Stab-QRAM: A Clifford-Only Quantum Oracle for Affine Boolean Data
提出Stab-QRAM编译器，将仿射布尔函数映射为仅含CNOT和X门的零T门电路，利用König边着色定理实现最小深度，适用于Simon型预言机、X型陪集算子的块编码及CSS码的综合征提取。
A universal set of quantum gates is constructed for the recently developed jump-error correcting quantum codes. These quantum codes are capable of correcting errors arising from the spontaneous decay of distinguishable qubits into statistically independent reservoirs. The proposed universal quantum gates are constructed with the help of Heisenberg- and Ising-type Hamiltonians acting on these physical qubits. This way it is guaranteed that the relevant error correcting code space is not left at any time even during the application of one of these quantum gates. The proposed entanglement gate is…
In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d is at least two. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a de…
Universality of a standard two-qubit gate by catalytic embedding
通过催化嵌套，研究发现受控-V门可实现通用量子计算，以最小开销模拟标准门集并提供合成算法。
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 find exact solutions for a universal set of quantum gates on a scalable candidate for quantum computers, namely an array of two level systems. The gates are constructed by a combination of dynamical and geometrical (non-Abelian) phases. Previously these gates have been constructed mostly on non-scalable systems and by numerical searches among the loops in the manifold of control parameters of the Hamiltonian.
We consider the implementation of two-qubit unitary transformations by means of CNOT gates and single-qubit unitary gates. We show, by means of an explicit quantum circuit, that together with local gates three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates.
It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of non-transversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here we demonstrate precisely the existence of such gates. In particular, we show how the limits of non-transversality can be overcome by performing rounds of intermediate error-correc…
Bernstein-Vazirani Algorithm with A CCNOT-Based Oracle
本研究提出一种基于Toffoli（CCNOT）逻辑门量子神谕的Bernstein-Vazirani问题量子算法，具备多项式加速特性，且可拓展求解新问题。
We present a nonlocal construction of universal gates by means of holonomic (geometric) quantum teleportation. The effect of the errors from imperfect control of the classical parameters, the looping variation of which builds up holonomic gates, is investigated. Additionally, the influence of quantum decoherence on holonomic teleportation used as a computational primitive is studied. Advantages of the holonomic implementation with respect to control errors and dissipation are presented.
In this paper is shown an application of Clifford algebras to the construction of computationally universal sets of quantum gates for $n$-qubit systems. It is based on the well-known application of Lie algebras together with the especially simple commutation law for Clifford algebras, which states that all basic elements either commute or anticommute.
Recently Shi proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a 'classical' set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev's universal set of gates, and discuss the meaning of the result.
What additional gates are needed for a set of classical universal gates to do universal quantum computation? We answer this question by proving that any single-qubit real gate suffices, except those that preserve the computational basis. The result of Gottesman and Knill[quant-ph/9807006] implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its alike) are universal f…
We numerically study the implementation of a universal two-qubit gate set, composed of CNOT, Hadamard, phase and $π/8$ gates, for transmon-based systems. The control signals to implement such gates are obtained using the Chopped Random Basis optimal control technique, with a target gate infidelity of $10^{-2}$. During the optimization processes we account for the leakage toward non-computational states, an important non-ideality affecting transmon qubits. We also test and benchmark the optimal control solutions against the introduction of Gaussian white noise and spectral distortion, two key n…
Realization of a Universal Quantum Gate Set for Itinerant Microwave Photons
本研究利用超导电路实现了微波频段无需后选择的确定性光子-光子通用门集合，测得单、双量子比特门保真度分别为75%和57%，为超导量子网络提供了新的技术路径。
We present some deterministic schemes to construct universal quantum gates, that is, controlled- NOT, three-qubit Toffoli, and Fredkin gates, between flying photon qubits and stationary electron-spin qubits assisted by quantum dots inside double-sided optical microcavities. The control qubit of our gates is encoded on the polarization of the moving single photon and the target qubits are encoded on the confined electron spins in quantum dots inside optical microcavities. Our schemes for these universal quantum gates on a hybrid system have some advantages. First, all the gates are accomplished…
We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.
We propose a scheme to realize universal quantum gates between nitrogen-vacancy (NV) centers in an optically trapped nanodiamond, through uniform magnetic field induced coupling between the NV centers and the torsional mode of the levitated nanodiamond. The gates are tolerant to the thermal noise of the torsional mode. By combining the scheme with dynamical decoupling technology, it is found that the high fidelity quantum gates are possible for the present experimental conditions. The proposed scheme is useful for NV-center-based quantum network and distributed quantum computation
The gate version of quantum computation exploits several quantum key resources as superposition and entanglement to reach an outstanding performance. In the way, this theory was constructed adopting certain supposed processes imitating classical computer gates. As for optical as well as magnetic systems, those gates are obtained as quantum evolutions. Despite, in certain cases they are attained as an asymptotic series of evolution effects. The current work exploits the direct sum of the evolution operator on a non-local basis for the driven bipartite Heisenberg-Ising model to construct a set o…
Stark shift on a superconducting qubit in circuit quantum electrodynamics (QED) has been used to construct universal quantum entangling gates on superconducting resonators in previous works. It is a second-order coupling effect between the resonator and the qubit in the dispersive regime, which leads to a slow state-selective rotation on the qubit. Here, we present two proposals to construct the fast universal quantum gates on superconducting resonators in a microwave-photon quantum processor composed of multiple superconducting resonators coupled to a superconducting transmon qutrit, that is,…
A large-scalable quantum computer model, whose qubits are represented by the subspace subtended by the ground state and the single exciton state on semiconductor quantum dots, is proposed. A universal set of quantum gates in this system may be achieved by a mixed approach, composed of dynamic evolution and nonadibatic geometric phase.
We demonstrate the applicability of a universal gate set in the parity encoding, which is a dual to the standard gate model, by exploring several quantum gate algorithms such as the quantum Fourier transform and quantum addition. Embedding these algorithms in the parity encoding reduces the circuit depth compared to conventional gate-based implementations while keeping the multiqubit gate counts comparable. We further propose simple implementations of multiqubit gates in tailored encodings and an efficient strategy to prepare graph states.
We propose a feasible scheme to implement a universal set of quantum gates based on geometric phases and superadiabatic quantum control. Consolidating the advantages of both strategies, the proposed quantum gates are robust and fast. The diamond nitrogen-vacancy center system is adopted as a typical example to illustrate the scheme. We show that these gates can be realized in a simple two-level configuration by appropriately controlling the amplitude, phase, and frequency of just one microwave field. The gate's robust and fast features are confirmed by comparing the fidelity of the proposed su…
A simultaneous realization of the Universal Optimal Quantum Cloning Machine (UOQCM) and of the Universal-NOT gate by a quantum injected optical parametric amplification (QIOPA), is reported. The two processes, forbidden in their exact form for fundamental quantum limitations, are found universal and optimal, and the measured fidelity F<1 is found close to the limit values evaluated by quantum theory. This work may enlighten the yet little explored interconnections of fundamental axiomatic properties within the deep structure of quantum mechanics.
This paper describes a novel approach to solving unstructured search problems using a classical, signal-based emulation of a quantum computer. The classical nature of the representation allows one to perform subspace projections in addition to the usual unitary gate operations. Although bandwidth requirements will limit the scale of problems that can be solved by this method, it can nevertheless provide a significant computational advantage for problems of limited size. In particular, we find that, for the same number of noisy oracle calls, the proposed subspace projection method provides a hi…
A single physical interaction might not be universal for quantum computation in general. It has been shown, however, that in some cases it can generate universal quantum computation over a subspace. For example, by encoding logical qubits into arrays of multiple physical qubits, a single isotropic or anisotropic exchange interaction can generate a universal logical gate-set. Recently, encoded universality for the exchange interaction was explicitly demonstrated on three-qubit arrays, the smallest nontrivial encoding. We now present the exact specification of a discrete universal logical gate-s…
Using Grover algorithm, this work investigates a technique for encoding search phrases used in wildcard searches. The technique involves creating a phase Oracle that loads data into a quantum circuit together with the search terms that have been encoded. The method entails constructing a phase Oracle programmatically using encoded input information and encoded search terms. By combining Grover diffusion operator with the phase Oracle, Hadamard gates, and zero-initialized three-qubit states, a complete quantum circuit is created. Trapped ion and superconducting qubit quantum computers, which we…
In order to achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable 'repeat-until-success' approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, although we require a primitive quantum memory.
Universal transversal gates
针对 Eastin-Knill 定理对通用横向门的限制，本文给出量子码具备通用横向门的充要条件，并构造可精确纠正局部与相关误差的码。
We show how one can implement any local quantum gate on specific qubits in an array of qubits by carrying adiabatically a Hamiltonian around a closed loop. We find the exact form of the loop and the Hamiltonian for implementing general one and two qubits gates. Our method is analytical and is not based on numerical search in the space of all loops.
We show how to construct quantum gate arrays that can be programmed to perform different unitary operations on a data register, depending on the input to some program register. It is shown that a universal quantum gate array - a gate array which can be programmed to perform any unitary operation - exists only if one allows the gate array to operate in a probabilistic fashion. The universal quantum gate array we construct requires an exponentially smaller number of gates than a classical universal gate array.
Many quantum programs require circuits for addition, subtraction and logical operations. These circuits may be packaged within routines known as oracles. However, oracles can be tedious to code with current frameworks. To solve this problem the author developed Higher-Level Oracle Description Language (HODL) -- a C-style programming language for use on quantum computers -- to ease the creation of such circuits. The compiler translates high-level code written in HODL and converts it into OpenQASM, a gate-based quantum assembly language that runs on IBM Quantum Systems and compatible simulators.
Many well-known and well-studied four by four universal quantum logic gates in the literature are of a specific form, the so called eight-vertex form \eqref{8vertexform} \cite{kaufman etal 05-1,kaufman etal 05-2}, or {\it similar} to it. We present a formalism for universal quantum logic gates of such a form. First, we provide explicit formulas in terms of matrix entries, which are the necessary and sufficient conditions for such a matrix to be a solution to the Yang-Baxter equation \eqref{byb}. Then, combining this with the conditions needed for being unitary \eqref{unitarycond} and being ent…
We propose a scalable scheme for optical quantum computing using measurement-induced continuous-variable quantum gates in a loop-based architecture. Here, time-bin-encoded quantum information in a single spatial mode is deterministically processed in a nested loop by an electrically programmable gate sequence. This architecture can process any input state and an arbitrary number of modes with almost minimum resources, and offers a universal gate set for both qubits and continuous variables. Furthermore, quantum computing can be performed fault-tolerantly by a known scheme for encoding a qubit …
Large optical nonlinearities occurring in a coherently prepared atomic system are shown to produce phase shifts of order $π$. Such an effect may be observed in ultracold rubidium atoms where it could be feasibly exploited toward the realization of a polarization phase gate.
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.
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