基于门的量子计算机通常以称为量子比特的二维单元来编码和处理信息。改用$d$维qudit可能带来内在优势，包括更高效的电路合成、问题定制的编码和嵌入式纠错。在本工作中，我们设计了一种基于超导qudit的量子处理器，其中transmon量子比特的逻辑空间被扩展到更高的激发能级。我们提出一个通用门集，包含一个双qudit交叉共振纠缠门，在$d=4$的ququart情形下，我们预测其保真度超过$99\%$，且采用现实实验参数。此外，我们提出一种分解例程，将一般qudit酉操作编译为这些基本门，所需的纠缠门数量少于量子比特替代方案。作为概念验证应用，我们数值演示了针对噪声量子硬件的${\ m SU}(16)$门合成，以及一个嵌入式纠错序列，该序列将量子比特存储器编码在transmon ququart中，以抵御纯退相位噪声。我们得出结论：通用qudit控制——作为超导量子信息处理操作工具箱的一个宝贵扩展——在当前基于transmon的架构中已可实现，并对近期和长期硬件都有应用价值。
英文摘要：
Gate-based quantum computers typically encode and process information in two-dimensional units called qubits. Using $d$-dimensional qudits instead may offer intrinsic advantages, including more efficient circuit synthesis, problem-tailored encodings and embedded error correction. In this work, we design a superconducting qudit-based quantum processor wherein the logical space of transmon qubits is extended to higher-excited levels. We propose a universal gate set featuring a two-qudit cross-resonance entangling gate, for which we predict fidelities beyond $99\%$ in the $d=4$ case of ququarts with realistic experimental parameters. Furthermore, we present a decomposition routine that compiles general qudit unitaries into these elementary gates, requiring fewer entangling gates than qubit alternatives. As proof-of-concept applications, we numerically demonstrate the synthesis of ${\rm SU}(16)$ gates for noisy quantum hardware and an embedded error correction sequence that encodes a qubit memory in a transmon ququart to protect against pure dephasing noise. We conclude that universal qudit control -- a valuable extension to the operational toolbox of superconducting quantum information processing -- is within reach of current transmon-based architectures and has applications to near-term and long-term hardware.
Based on a microwave-photon quantum processor with two superconducting resonators coupled to one transmon qutrit, we construct the controlled-phase (c-phase) gate on microwave-photon-resonator qudits, by combination of the photon-number-dependent frequency-shift effect on the transmon qutrit by the first resonator and the resonant operation between the qutrit and the second resonator. This distinct feature provides us a useful way to achieve the c-phase gate on the two resonator qudits with a higher fidelity and a shorter operation time, compared with the previous proposals. The fidelity of ou…
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…
Experiments with superconducting circuits require careful calibration of the applied pulses and fields over a large frequency range. This remains an ongoing challenge as commercial semiconductor electronics are not able to probe signals arriving at the chip due to its cryogenic environment. Here, we demonstrate how the on-chip amplitude and frequency of a microwave signal can be inferred from the ac Stark shifts of higher transmon levels. In our time-resolved measurements we employ Ramsey fringes, allowing us to detect the amplitude of the systems transfer function over a range of several hund…
We propose a linear-size synthesis of the multi-controlled Toffoli gate on qudits with at most one borrowed ancilla. This one ancilla can even be saved when the qudit dimension is odd. Our synthesis leads to improvements in various quantum algorithms implemented on qudits. In particular, we obtain (i) a linear-size and one-clean-ancilla synthesis of multi-controlled qudit gates; (ii) an optimal-size and one-clean-ancilla synthesis of unitaries on qudits; (iii) a near-optimal-size and ancilla-free/one-borrowed-ancilla implementation of classical reversible functions as qudit gates.
We introduce quantum hybrid gates that act on qudits of different dimensions. In particular, we develop two representative two-qudit hybrid gates (SUM and SWAP) and many-qudit hybrid Toffoli and Fredkin gates. We apply the hybrid SUM gate to generating entanglement, and find that operator entanglement of the SUM gate is equal to the entanglement generated by it for certain initial states. We also show that the hybrid SUM gate acts as an automorphism on the Pauli group for two qudits of different dimension under certain conditions. Finally, we describe a physical realization of these hybrid gat…
We experimentally constructed an all-microwave scheme for the controlled-NOT (cNOT) gate between two superconducting transmon qubits in a three dimensional cavity. Our cNOT gate is based on the microwave-activated phase (MAP) gate, which requires an additional procedure to compensate the accumulated phases during the operation of the MAP gate. We applied Z-axis phase gates using microwave hyperbolic secant pulse on both qubits with adequate rotation angles systematically calibrated by separate measurements.We evaluated the gate performance of the constructed cNOT gate by performing two-qubit q…
We present a scalable set of universal gates and multiply controlled gates in a qudit basis through a bijective mapping from N qubits to qudits with D = 2^N levels via rotations in U(2). For each of the universal gates (H, CNOT, and T), as well as the NOT gate and multiply-controlled-Z gates, we describe a systematic approach to identifying the set of U(2) rotations required to implement each gate for any qudit of size N and with minimal use of an ancilla level. The qudit gates are analyzed in terms of the total rotation count and gate depth as the system scales with D. We apply the qudit-basi…
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…
We present a quantum SWAP gate valid for quantum systems of an arbitrary dimension. The gate generalizes the CNOT implementation of the SWAP gate for qubits and keeps its most important properties, like symmetry and simplicity. We only use three copies of the same controlled qudit gate. This gate can be built with two standard higher-dimensional operations, the Quantum Fourier Transform and the d-dimensional version of the CZ gate.
We propose three core ideas: 1. the wave-particle duality of the qudit quantum space; 2. the classification of all elementary quantum gates by ordered pairs of qudit functionals; 3. a new type of quantum gates called the "quantum wave gates". We first study the quantum functionals whose relation to the quantum states is analogous to that between the momentum and position wavefunctions in fundamental quantum physics: a Fourier transform and an entropic uncertainty principle can be defined between the dual representations. The quantum functionals are not just mathematical constructs but have cle…
We discuss the implementation of quantum algorithms for lattice $Φ^4$ theory on circuit quantum electrodynamics (cQED) system. The field is represented on qudits in a discretized field amplitude basis. The main advantage of qudit systems is that its multi-level characteristic allows the field interaction to be implemented only with diagonal single-qudit gates. Considering the set of universal gates formed by the single-qudit phase gate and the displacement gate, we address initial state preparation and single-qudit gate synthesis with variational methods.
We have embedded two fixed-frequency Al/AlO$_{\textrm{x}}$/Al transmons, with ground-to-excited transition frequencies at 6.0714 GHz and 6.7543 GHz, in a single 3D Al cavity with a fundamental mode at 7.7463 GHz. Strong coupling between the cavity and each transmon results in an effective qubit-qubit coupling strength of 26 MHz and a -1 MHz dispersive shift in each qubit's transition frequency, depending on the state of the other qubit. Using the all-microwave SWIPHT (Speeding up Waveforms by Inducing Phases to Harmful Transitions) technique, we demonstrate the operation of a generalized contr…
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）
We concretely construct an extension of the controlled-U gate in qudit from some elementary gates. We also construct unitary transformation in two-qudit by means of the extended controlled-U gate and show the universality of it.
The application of high-dimensional quantum systems (qudits) in quantum computing and communications seems to be a promising avenue due to the possibility of increasing the amount of information encoded in one physical carrier. In this work, we propose a method for implementing single-qudit gates for qudits based on light modes with orbital angular momentum. Method for logical qudits encoding, which ensures the quasi-cyclicity of operations, is introduced. Based on the protocol for converting the orbital angular momentum of light in the Raman quantum memory scheme [Vashukevich E.A. et. al. PRA…
Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The imp…
High-fidelity iSWAP gate with Double Transmon Coupler
University of Massachusetts, Lowell（马萨诸塞大学洛厄尔分校）；RTX BBN Technologies（RTX BBN科技公司）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Air Force Research Laboratory（空军研究实验室）；Northwestern University（西北大学）
Cross-Resonant Gates in Hybrid Fluxonium-Transmon Systems
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；SEEQC, Inc.（SEEQC公司）；University of Massachusetts-Amherst（马萨诸塞大学阿默斯特分校）；Ecole Polytechnique Federale de Lausanne（洛桑联邦理工学院）
We present quantum networks for a n-qubit controlled gate C^{n-1}(U) which use a higher dimensional (qudit) ancilla as a catalyser. In its simplest form the network has only n two-particle gates (qubit-qudit) -- this is the minimum number of two-body interactions needed to couple all n+1 subsystems (n qubits plus one ancilla). This class of controlled gates includes the generalised Toffoli gate C^{n-1}(X) on n qubits, which plays an important role in several quantum algorithms and error correction. A particular example implementing this model is given by the dispersive limit of a generalised J…
We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, which may be interesting from the experimental point of view. These preparations can be seen as the outcomes of certain minimal positive operator valued measures. The connection of our results with these generalized measurements is briefly discussed.
Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP g…
This paper concerns the efficient implementation of quantum circuits for qudits. We show that controlled two-qudit gates can be implemented without ancillas and prove that the gate library containing arbitrary local unitaries and one two-qudit gate, CINC, is exact-universal. A recent paper (PRL 94 230502) describes quantum circuits for qudits which require O(d^n) two-qudit gates for state synthesis and O(d^{2n}) two-qudit gates for unitary synthesis, matching the respective lower bound complexities. In this work, we present the state synthesis circuit in much greater detail and prove that it i…
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…
Parametrically Driven iSWAP Gate Using a Capacitively Shunted Double-Transmon Coupler at the Zero-Flux Sweet Spot
RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；Department of Applied Physics, Graduate School of Engineering, The University of Tokyo（东京大学大学院工学研究科应用物理系）；Corporate Laboratory, Toshiba Corporation（东芝公司企业实验室）
Quantum computation requires the precise control of the evolution of a quantum system, typically through application of discrete quantum logic gates on a set of qubits. Here, we use the cross-resonance interaction to implement a gate between two superconducting transmon qubits with a direct static dispersive coupling. We demonstrate a practical calibration procedure for the optimization of the gate, combining continuous and repeated-gate Hamiltonian tomography with step-wise reduction of dominant two-qubit coherent errors through mapping to microwave control parameters. We show experimentally …
A qudit ($d$-level quantum systems) has a large Hilbert space and thus can be used to achieve many quantum information and communication tasks. Here, we propose a method to transfer arbitrary $d$-dimensional quantum states (known or unknown) between two superconducting qudits coupled to a single cavity. The state transfer can be performed fast because of employing resonant interactions only. In addition, quantum states can be deterministically transferred without measurement. Numerical simulations show that high-fidelity transfer of quantum states between two superconducting transmon qudits ($…
This paper presents a highly efficient decomposition scheme and its associated Mathematica notebook for the analysis of complicated quantum circuits comprised of single/multiple qubit and qudit quantum gates. In particular, this scheme reduces the evaluation of multiple unitary gate operations with many conditionals to just two matrix additions, regardless of the number of conditionals or gate dimensions. This improves significantly the capability of a quantum circuit analyser implemented in a classical computer. This is also the first efficient quantum circuit analyser to include qudit quantu…
Efficient High-Dimensional Quantum Circuit Synthesis: From Multi-Controlled Gates to Isometries and Quantum Channels
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）；School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）
Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. Advantages of hybrid and d-valued gates (circuits) and their physical realizations have been studied in detail by Muthukrishnan and Stroud (Physical Review A, 052309, 2000), Daboul et al. (J. Phys. A: Math. Gen. 36 2525-2536, 2003), and Bartlett et al (Physical Review A, Vol.65, 052316, 2002). In both cases, a …
Experimental Realization of a C-Shunt Flux-Qubit Coupler for Controlled-Z Gates between Transmons
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所凝聚态物理国家实验室）；School of Physics, University of Chinese Academy of Sciences（中国科学院大学物理学院）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）
The SWAP gate has become an integral feature of quantum circuit architectures and is designed to permute the states of two qubits through the use of the well-known controlled-NOT gate. We consider the question of whether a two-qudit quantum circuit composed entirely from instances of the generalised controlled-NOT gate can be constructed to permute the states of two qudits. Arguing via the signature of a permutation, we demonstrate the impossibility of such circuits for dimensions $d \equiv 3$ (mod 4).
Fast microwave-driven two-qubit gates between fluxonium qubits with a transmon coupler
QuTech and Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学量子技术与卡弗里纳米科学研究所）；Netherlands Organization for Applied Scientific Research (TNO)（荷兰应用科学研究组织（TNO））；Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol（布里斯托大学量子工程博士培训中心、H.H.威尔斯物理实验室及电气与电子工程系）
We propose the generalized controlled X (GCX) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Sha…
