CNOT Research Articles

Harnessing Nth Root Gates for Energy Storage

We explore the use of fractional controlled-not gates in quantum thermodynamics. The Nth-root gate allows for a paced application of two-qubit operations. We apply it in quantum thermodynamic protocols for charging a quantum battery. Circuits for three (and two) qubits are analysed by considering the generated ergotropy and other measures of performance. We also perform an optimisation of initial system parameters, e.g.,the initial quantum coherence of one of the qubits strongly affects the efficiency of protocols and the system’s performance as a battery. Finally, we briefly discuss the feasibility for an experimental realization.

Quantum circuit syntheses of Boolean matrix multiplication using only CNOT gates

Abstract Boolean matrices and operations on Boolean matrices have become increasingly popular in quantum computing and quantum information because they can be used to represent quantum states and the corresponding operations. In this paper, we first give a quantum circuit direct synthesis of general Boolean matrix multiplication using Toffoli gates. We assign a qubit to each element of the matrix and add Toffoli gates at the corresponding positions where the matrix elements are multiplied according to the arithmetic of matrix multiplication. For $n\times p$ dimensional matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $np + pm + nm$ qubits and at most $pmn$ Toffoli gates are used in total. Then, an improved heuristic algorithm based on matrix Hamming distance selection of CNOT gate is proposed to synthesize the quantum circuit of a matrix, without using wire switching and SWAP gates, using only CNOT gates. The results show that our method is as good as or better than the classical synthesis. Finally, the quantum circuit of Boolean matrix multiplication with at least one invertible or full-rank matrix using only CNOT gates is given, which is divided into five cases for discussion and analysis. For $n\times n$ dimensional invertible matrix $A$ multiplied by $n\times m$ dimensional matrix $B$, $nm$ qubits and at most $mn^{2}$ CNOT gates are used in total. For $n\times p$ dimensional column full-rank matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $nm$ qubits and at most $pmn$ CNOT gates are used in total. The results show that this method uses less quantum resources to synthesize the quantum circuit of Boolean matrix multiplication. The correctness of the quantum circuits was verified by the Aer simulator of the IBM Quantum platform.

Pseudo twirling mitigation of coherent errors in non-Clifford gates

The conventional circuit paradigm, utilizing a small set of gates to construct arbitrary quantum circuits, is hindered by significant noise. In the quantum Fourier transform, for instance, the standard gate paradigm employs two CNOT gates for the partial CPhase. In contrast, some quantum computers can directly implement such operations using their native interaction, resulting in less noisy gates. Unfortunately, coherent errors degrade the performance of these gates. In Clifford gates such as the CNOT, these errors can be addressed through randomized compiling (RC). However, RC does not apply to the non-Clifford multi-qubit native implementations described above. The present work introduces and experimentally demonstrates a technique called ‘Pseudo Twirling’ (PST) to address coherent errors. We demonstrate experimentally that integrating PST with the ‘Adaptive KIK’ quantum error mitigation method enables the simultaneous mitigation of noise and coherent errors in multi-qubit non-Clifford gates.

Towards High Performance QNNs via Distribution-Based CNOT Gate Reduction

Quantum Neural Networks (QNNs) are one of the most promising applications that can be implemented on NISQ-era quantum computers. In this study, we observe that QNNs often suffer from gate redundancy, which hugely declines the performance and accuracy of the network. Even state-of-the-art architecture search techniques like QuantumNAS do not completely alleviate this problem. Especially, We find that CNOT gates are major contributors to the execution delay and noise in quantum circuits, and there are many redundant CNOT gates in the QNN post-training. This motivates us to propose a novel distribution-based greedy-search circuit optimization technique, that can be employed after the completion of the training process. Our technique significantly reduces the number of CNOT gates in QNNs without affecting the accuracy of the network. With this technique, we have achieved an average of 3 × improvement in execution time while reaching a maximum of 12.4 × improvement.

Implementation of controlled-not quantum gate by nonlinear coupled electro-nano-optomechanical oscillators

Implementation of controlled-not quantum gate by nonlinear coupled electro-nano-optomechanical oscillators

Linear circuit synthesis using weighted Steiner trees

CNOT circuits are a common building block of general quantum circuits. The problem of synthesizing and optimizing such circuits has received a lot of attention in the quantum computing literature. This problem is especially challenging for quantum devices with restricted connectivity, where two-qubit gates can only be placed between adjacent qubits. The state-of-the-art algorithms for optimizing the number of CNOT gates are heuristic algorithms that are based on Gaussian elimination and that use Steiner trees to connect between different subsets of qubits. In this article, we suggest considering \emph{weighted} Steiner trees, and we present a simple low-cost heuristic to compute weights. The simulated evaluation shows that the suggested heuristic is almost always beneficial and reduces the number of CNOT gates by up to 10%.

Delayed-measurement one-way quantum computing on cloud quantum computer

Abstract One-way quantum computation focuses on initially generating an entangled cluster state followed by a sequence of measurements with classical communication of their individual outcomes. Recently, a delayed-measurement approach has been applied to replace classical communication of individual measurement outcomes. In this work, by considering the delayed-measurement approach, we demonstrate a modified one-way CNOT gate using the on-cloud superconducting quantum computing platform: Quafu. The modified protocol for one-way quantum computing requires only three qubits rather than the four used in the standard protocol. Since this modified cluster state decreases the number of physical qubits required to implement one-way computation, both the scalability and complexity of the computing process are improved. Compared to previous work, this modified one-way CNOT gate is superior to the standard one in both fidelity and resource requirements. We have also numerically compared the behavior of standard and modified methods in large-scale one-way quantum computing. Our results suggest that in a noisy intermediate-scale quantum (NISQ) era, the modified method shows a significant advantage for one-way quantum computation.

A Multiparty Quantum Private Equality Comparison Scheme Relying on |GHZ3⟩ States

In this work, we present a new protocol that accomplishes multiparty quantum private comparison leveraging maximally entangled |GHZ3⟩ triplets. Our intention was to develop a protocol that can be readily executed by contemporary quantum computers. This is possible because the protocol uses only |GHZ3⟩ triplets, irrespective of the number n of millionaires. Although it is feasible to prepare multiparticle entangled states of high complexity, this is overly demanding on a contemporary quantum apparatus, especially in situations involving multiple entities. By relying exclusively on |GHZ3⟩ states, we avoid these drawbacks and take a decisive step toward the practical implementation of the protocol. An important quantitative characteristic of the protocol is that the required quantum resources are linear both in the number of millionaires and the amount of information to be compared. Additionally, our protocol is suitable for both parallel and sequential execution. Ideally, its execution is envisioned to take place in parallel. Nonetheless, it is also possible to be implemented sequentially if the quantum resources are insufficient. Notably, our protocol involves two third parties, as opposed to a single third party in the majority of similar protocols. Trent, commonly featured in previous multiparty protocols, is now accompanied by Sophia. This dual setup allows for the simultaneous processing of all n millionaires’ fortunes. The new protocol does not rely on a quantum signature scheme or pre-shared keys, reducing complexity and cost. Implementation wise, uniformity is ensured as all millionaires use similar private circuits composed of Hadamard and CNOT gates. Lastly, the protocol is information-theoretically secure, preventing outside parties from learning about fortunes or inside players from knowing each other’s secret numbers.

Multipass quantum process tomography

We introduce a method to enhance the precision and accuracy of Quantum Process Tomography (QPT) by mitigating the errors caused by state preparation and measurement (SPAM), readout and shot noise. Instead of performing QPT solely on a single gate, we propose performing QPT on a sequence of multiple applications of the same gate. The method involves the measurement of the Pauli transfer matrix (PTM) by standard QPT of the multipass process, and then deduce the single-process PTM by two alternative approaches: an iterative approach which in theory delivers the exact result for small errors, and a linearized approach based on solving the Sylvester equation. We examine the efficiency of these two approaches through simulations on IBM Quantum using ibmq_qasm_simulator. Compared to the Randomized Benchmarking type of methods, the proposed method delivers the entire PTM rather than a single number (fidelity). Compared to standard QPT, our method delivers PTM with much higher accuracy and precision because it greatly reduces the SPAM, readout and shot noise errors. We use the proposed method to experimentally determine the PTM and the fidelity of the CNOT gate on the quantum processor ibmq_manila (Falcon r5.11L).

Theoretical concept of error-rejecting entanglement purification of quantum-dots electronic spins in single-sided optical microcavity

We use the quantum-dots (QD) coupled with single-sided microcavities system to construct the error-rejecting controlled not (CNOT) gate of electronic spins system in QD; We have designed an error-rejecting entanglement purification protocol (EPP) of electronic spins system in QD by using the error-rejecting CNOT gate and unitary operation. It can extract the high entanglement from the mixed entanglement states of the electronic spins with low entangled states. This EPP can eliminate the operation errors caused by the nonideal interaction between photons and QD coupled with optical microcavities system, and improve the fidelity of the EPP through iteration. Our scheme is more practical in future long-distance quantum communications, especially providing significant benefits for solving decoherence problems in quantum networks and quantum repeaters.

Building a controlled-NOT gate between polarization and frequency

Building a controlled-NOT gate between polarization and frequency

Performance of entanglement purification including maximally entangled mixed states

Entanglement between distant quantum systems is a critical resource for implementing quantum communication. This property is affected by external agents and can be restored by employing efficient entanglement purification protocols. In this work, we propose an entanglement purification protocol based on two entangling two-qubit operations that replace the usual controlled-NOT (CNOT) gate. These operations arise from a generalised quantum measurement and can be understood as measurement operators in a positive operator-valued measure. Furthermore, two variants of the core protocol are introduced and their performances are studied in terms of the overall success probability of reaching a Bell state and the number of purifiable states. Based on rank-two states, we can obtain analytical expressions for the success probability that we extend and refine using numerical calculations to the case of maximally entangled states. We also consider more general rank-three states to show that our procedure is in general more convenient compared to purification protocols based on Bell diagonal states. Finally, we test the protocols using initial random states. In all cases, we find a better performance and a larger amount of purifiable states using our schemes compared to the CNOT-based protocol.

A deterministic qudit-dependent high-dimensional photonic quantum gate with weak cross-kerr nonlinearity

High-dimensional quantum systems expand the Hilbert space, enabling the processing of more intricate information and the execution of wider quantum operations. The development of high-dimensional quantum systems relies significantly on the implementation of high-dimensional gates. In the paper, we propose a deterministic 4 × 4-dimensional controlled-not (CNOT) gate for a three-photon system, where encoding on the polarization-spatial state of a single photon acts as a 4-dimensional control qudit, meanwhile the polarized state of the remaining two photons serves as a 4-dimensional target qudit. The implementation of the gate leverages weak cross-kerr nonlinearities and X-homodyne detectors, which impose lower requirements on the interaction strength and demonstrate robustness against photon loss. In accordance with the measurement of the coherent state, the relevant classical feed-forward operations are employed to make high-dimensional CNOT gate effective. Additionally, the proposed scheme demonstrates a high successful probability as a result of the utilization of mature measurement methods and straightforward optical elements. Further, the fidelity of the CNOT gate is robust against photon loss and is approximate to 1 with the existing technology. Consequently, our deterministic protocol presents a promising approach for the practical realization of high-dimensional quantum computation for photon systems.

Polylogarithmic-depth controlled-NOT gates without ancilla qubits

Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n-control-NOT gates (Cn(X)) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces Cn(X) circuits outperforming previous methods in the asymptotic and non-asymptotic regimes. Three distinct decompositions are presented: an exact one using one borrowed ancilla with a circuit depth Θ(log(n)3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta (\\log {(n)}^{3})$$\\end{document}, an approximating one without ancilla qubits with a circuit depth O(log(n)3log(1/ϵ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}(\\log {(n)}^{3}\\log (1/\\epsilon ))$$\\end{document} and an exact one with an adjustable-depth circuit which decreases with the number m≤n of ancilla qubits available as O(log(n/⌊m/2⌋)3+log(⌊m/2⌋))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{O}}}}}}}}(\\log {(n/\\lfloor m/2\\rfloor )}^{3}+\\log (\\lfloor m/2\\rfloor ))$$\\end{document}. The resulting exponential speedup is likely to have a substantial impact on fault-tolerant quantum computing by improving the complexities of countless quantum algorithms with applications ranging from quantum chemistry to physics, finance and quantum machine learning.

Efficient entanglement enhancement of partially entangled pairs

We present a comprehensive approach to transform a partially entangled pure state into a maximally entangled state, utilizing only local operations and a single copy of the entangled state. Our approach eliminates the need for multiple copies of entangled states, bilateral operations, and the time-consuming partial-collapse weak measurements. Integrating ancilla qubits, rotation gates, and CNOT gates, we simplify the process of enhancing entanglement, achieving significant efficiency using only a single partially entangled state. Assuming perfect local operations, we showcase the efficacy of our method through both mathematical analysis and the practical implementation of the corresponding circuit in Qiskit. Furthermore, we explore the impact of imperfect local operations, specifically focusing on the CNOT gate and measurement, for a comprehensive analysis. Additionally, we address the case of unknown parameters in the partially entangled pure state and introduce a robust method to effectively handle this uncertainty, ensuring reliable entanglement fidelity optimization in practical situations. We extend our investigation to the application of our method to mixed initial entangled states. Our results demonstrate the reliability of our method in increasing entanglement even in the presence of phase damping and amplitude damping. Additionally, we illustrate its effectiveness in transforming specific mixed initial states in the presence of depolarization into states with enhanced entanglement.

Controlled remote implementation of operations for many systems

Quantum communication plays a key role in the next generation of information transfer and security schemes, by using quantum entanglement and measurements from quantum mechanics. We introduce the mathematical and physical foundations of quantum communication, such as the CNOT gate and Kronecker product. Then we propose two quantum communication protocols, namely dense coding and stealth coding. These protocols offer unique advantages that are theoretically unbreakable. Then we extend the protocols to the quantum communication protocol allowing for third-party supervision to ensure information security. Based on this, a communication protocol involving four parties is designed, enabling them to exchange information while being supervised.

Simplified quantum teleportation for a distinctive six-qubit target state via a six-qubit entangled state

Based on a six-qubit entangled state, a quantum information processing scheme for teleporting a distinctive six-qubit state is presented. In the scheme, only Bell-state measurements and two-qubit controlled-NOT gate operations as well as some single-qubit transformed operations are needed. Compared with a rival scheme put forwarded by Tan et al [Int. J. Theor. Phys. 55, 155 (2016)], the present scheme is more simpler and easier to execute because it does not require to make the six-qubit entangled state measurement. Besides, it is deterministic and feasible in terms of the current experimental technologies.

Reinforcement learning pulses for transmon qubit entangling gates

The utility of a quantum computer is highly dependent on the ability to reliably perform accurate quantum logic operations. For finding optimal control solutions, it is of particular interest to explore model-free approaches, since their quality is not constrained by the limited accuracy of theoretical models for the quantum processor—in contrast to many established gate implementation strategies. In this work, we utilize a continuous control reinforcement learning algorithm to design entangling two-qubit gates for superconducting qubits; specifically, our agent constructs cross-resonance and CNOT gates without any prior information about the physical system. Using a simulated environment of fixed-frequency fixed-coupling transmon qubits, we demonstrate the capability to generate novel pulse sequences that outperform the standard cross-resonance gates in both fidelity and gate duration, while maintaining a comparable susceptibility to stochastic unitary noise. We further showcase an augmentation in training and input information that allows our agent to adapt its pulse design abilities to drifting hardware characteristics, importantly, with little to no additional optimization. Our results exhibit clearly the advantages of unbiased adaptive-feedback learning-based optimization methods for transmon gate design.

Enhancing scalability and accuracy of quantum poisson solver

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem and thus have no practical usage. In this regard, our previous work (Robson in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), 2022) showed a proof-of-concept demonstration in advancing quantum Poisson solver algorithm and validated preliminary results for a simple case of 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3\ imes 3$$\\end{document} problem. In this work, we delve into comprehensive research details, presenting the results on up to 15×15\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$15\ imes 15$$\\end{document} problems that include step-by-step improvements in Poisson equation solutions, scaling performance, and experimental exploration. In particular, we demonstrate the implementation of eigenvalue amplification by a factor of up to 28\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^8$$\\end{document}, achieving a significant improvement in the accuracy of our quantum Poisson solver and comparing that to the exact solution. Additionally, we present success probability results, highlighting the reliability of our quantum Poisson solver. Moreover, we explore the scaling performance of our algorithm against the circuit depth and width, demonstrating how our approach scales with larger problem sizes and thus further solidifies the practicality of easy adaptation of this algorithm in real-world applications. We also discuss a multilevel strategy for how this algorithm might be further improved to explore much larger problems with greater performance. Finally, through our experiments on the IBM quantum hardware, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.

Single-step parity check gate set for quantum error correction

A key requirement for an effective quantum error correction (QEC) scheme is that the physical qubits have error rates below a certain threshold. The value of this threshold depends on the details of the specific QEC scheme, and its hardware-level implementation. This is especially important with parity-check circuits, which are the fundamental building blocks of QEC codes. The standard way of constructing the parity check circuit is using a universal set of gates, namely sequential CNOT gates, single-qubit rotations and measurements. We exploit the insight that a QEC code does not require universal logic gates, but can be simplified to perform the sole task of error detection and correction. By building gates that are fundamental to QEC, we can boost the threshold and ease the experimental demands on the physical hardware. We present a rigorous formalism for constructing and verifying the error behavior of these gates, linking the physical measurement of a process matrix to the abstract error models commonly used in QEC analysis. This allows experimentalists to directly map the gates used in their systems to thresholds derived for a broad-class of QEC codes. We give an example of these new constructions using the model system of two nuclear spins, coupled to an electron spin, showing the potential benefits of redesigning fundamental gate sets using QEC primitives, rather than traditional gate sets reliant on simple single and two-qubit gates.