Toffoli Gate Research Articles

Post-selection-free preparation of high-quality physical qubits

Rapidly improving gate fidelities for coherent operations mean that errors in state preparation and measurement (SPAM) may become a dominant source of error for fault-tolerant operation of quantum computers. This is particularly acute in superconducting systems, where tradeoffs in measurement fidelity and qubit lifetimes have limited overall performance. Fortunately, the essentially classical nature of preparation and measurement enables a wide variety of techniques for improving quality using auxiliary qubits combined with classical control and post-selection. In practice, however, post-selection greatly complicates the scheduling of processes such as syndrome extraction. Here we present a family of quantum circuits that prepare high-quality |0⟩ states without post-selection, instead using CNOT and Toffoli gates to non-linearly permute the computational basis. We find meaningful performance enhancements when two-qubit gate fidelities errors go below 0.2%, and even better performance when native Toffoli gates are available.

Development of a quantum calculator on the IBM Quantum experience platform and its application for converting the decimal number system to binary

This article demonstrates a quantum calculator by experimentally simulating and performing basic arithmetic operations, namely addition, subtraction, multiplication, division on real quantum devices using the IBM Quantum Experience (IBM QE) platform. Two addition algorithms are considered, the classical version, using the Toffoli gate, and using the quantum Fourier transform (QFT). In particular summation with QFT saves memory and time. The task of the work is to build a quantum circuit that performs arithmetic operations between the binary representation of two integers. We then propose a quantum algorithm for converting decimal to binary. This method also allows for massive parallelization during its execution. The addition, subtraction, multiplication, and division functions based on QFT have been improved with some changes. The proposed operations are compared with the closest quantum arithmetic operations. Therefore, we can implement a practical quantum calculator by performing operations on a real quantum computer. The circuits are defined as a kind of quantum adder using a quantum Fourier transform, implemented in Qiskit and demonstrated on one or two copies.

Optical multiqubit gate operations on an excitation-blockaded atomic quantum register

We consider a multiqubit system of atoms or ions with two computational ground states and an interacting excited state in the so-called blockade regime, such that only one qubit can be excited at any one time. Examples of such systems are rare-earth-ion-doped crystals and neutral atoms trapped in tweezer arrays. We present a simple laser excitation protocol that yields a complex phase factor on any desired multiqubit product state, and which can be used to implement multiqubit gates such as the $n$-bit Toffoli gates. The operation is performed using only two pulses, where each pulse simultaneously address all qubits involved in the operation. By relying on the well-established complex hyperbolic secant pulse shape, our scheme is also robust and permits complete transfers to and from the excited states despite variations in Rabi frequencies and transition frequencies. A detailed analysis of the multiqubit gate performance is provided.

Linear Optical Universal Quantum Gates with Higher Success Probabilities

AbstractUniversal quantum gates lie at the heart of designing quantum computer. Here two compact quantum circuits for implementing post‐selected controlled‐phase‐flip (CPF) gate and Toffoli gate with linear optics are constructed 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, the presented Toffoli gate is accomplished with current maximum success probability of 1/30 and unity fidelity in principle, without using additional entangled photon pairs and the standard decomposition‐based approach. Linear optical implementations of the presented two universal gates are feasible and economical under current technology, and provide a potential application in large‐scale optical quantum computing.

A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate

Set Intersection Cardinality (SI-CA) computes the intersection cardinality of two parties' sets, which has many important and practical applications such as data mining and data analysis. However, in the face of big data sets, it is difficult for two parties to execute the SI-CA protocol repeatedly. In order to reduce the execution pressure, a Private Set Intersection Cardinality (PSI-CA) protocol based on a quantum homomorphic encryption scheme for the Toffoli gate is proposed. Two parties encode their private sets into two quantum sequences and encrypt their sequences by way of a quantum homomorphic encryption scheme. After receiving the encrypted results, the semi-honest third party (TP) can determine the equality of two quantum sequences with the Toffoli gate and decrypted keys. The simulation of the quantum homomorphic encryption scheme for the Toffoli gate on two quantum bits is given by the IBM Quantum Experience platform. The simulation results show that the scheme can also realize the corresponding function on two quantum sequences.

Quantum Binary Field Multiplication with Optimized Toffoli Depth and Extension to Quantum Inversion

The Shor's algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor's algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh-Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).

Effective implementation of nonadiabatic geometric quantum gates of cat-state qubits using an auxiliary qutrit

We propose an effective protocol for the implementation of nonadiabatic geometric quantum gates of cat-state qubits in Kerr-nonlinear resonators driven by two-photon squeezing drives. Coupling the Kerr-nonlinear resonators with an auxiliary qutrit with proper coupling strengths, the selective transition of the auxiliary qutrit is realized. The selective transition can be exploited in the implementation of a set of useful quantum gates, including the phase gates, the NOT gates, the controlled-phase gates, the controlled NOT gates, and the Toffoli gates. Numerical simulations show the implementations of different types of gates are robust against systematic errors, random noise, and decoherence. Therefore, the protocol may be helpful for robust and scalable quantum computation based on cat-state qubits.

An efficient design and implementation of a reversible logic CCNOT (Toffoli) gate in QCA for nanotechnology

An efficient design and implementation of a reversible logic CCNOT (Toffoli) gate in QCA for nanotechnology

Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover's Algorithm with Qudits.

Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states-qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems-so-called ququints-that use ququints' space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover's algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

A QND-based Toffoli gate for polarization qubits and its applications

A QND-based Toffoli gate for polarization qubits and its applications

An Efficient Method to Decompose and Map MPMCT Gates That Accounts for Qubit Placement

Mixed-Polarity Multiple-Control Toffoli (MPMCT) gates are generally used to implement large control logic functions for quantum computation. A logic circuit consisting of MPMCT gates needs to be mapped to a quantum computing device that invariably has a physical limitation, which means we need to (1) decompose the MPMCT gates into one- or two-qubit gates, and then (2) insert SWAP gates so that all the gates can be performed on Nearest Neighbor Architectures (NNAs). Up to date, the above two processes have only been studied independently. In this work, we investigate that the total number of gates in a circuit can be decreased if the above two processes are considered simultaneously as a single step. We developed a method that inserts SWAP gates while decomposing MPMCT gates unlike most of the existing methods. Also, we consider the effect on the latter part of a circuit carefully by considering the qubit placement when decomposing an MPMCT gate. Experimental results demonstrate the effectiveness of our method.

The Fredkin gate in reversible and quantum environments

Reversible Computing circuits are characterized by low power consumption and their proximity to circuits for quantum computing. The Fredkin gate was one of the earliest proposed controlled reversible circuits, which however, was soon superseded by the Toffoli gate, the NOT, and CNOT gates, which constituting a flexible functionally complete set could also realize the Fredkin gate as a building block. In quantum computing circuits, the Fredkin gate (under the name controlled SWAP) plays an important role regarding the superposition of states. The present paper studies extensions of the Fredkin gate in terms of mixed polarity in the reversible domain and an application in quantum computing.

Optical scheme of developing Hamming code-based error detection and correction using Toffoli gates with frequency encoded data

Bit-error detection and correction is a very important part of a data communication network. In this article, we propose an optical approach to designing an error detection and correction circuit using Hamming code-based frequency encoded binary data and Toffoli gates. Here, the Toffoli gates are made of semiconductor optical amplifiers (SOAs) that work based on the nonlinear polarization rotation property. The characteristics of SOA, including very fast switching action (about 0.01 ns), low power consumption, frequency independent cross-polarization gain in the C-band, cascading property, etc., make the scheme attractive. The frequency encoded data-based proposed scheme will be effective for WDM communication networks. MATLAB simulation results testify to the feasibility of the proposed scheme.

Pockel’s effect inspired Toffoli Gate: an MZI count optimized design and logical applications

Pockel’s effect inspired Toffoli Gate: an MZI count optimized design and logical applications

Hyper-parallel nonlocal Toffoli gates assisted by quantum dots inside a double-side optical microcavity

<p indent="0mm">Implementing three-qubit Toffoli gates nonlocally plays an important role in quantum communication networks and distributed quantum computation. Most of previous protocols for implementing quantum Toffolis gate consider only implementing Toffoli gates with one degree of freedom. In this paper, we present a scheme for parallel implementation of nonlocal Toffoli gates in polarization and spatial-mode degrees of freedom assisted by quantum dots inside a double-side optical microcavity. Alice and Bob let photons traverse the double-side optical microcavity sequentially and perform single-particle measurements on electron spin in quantum dots and the entangled photons. Charlie performs single-particle measurements and applies the corresponding unitary operation in polarization and spatial-mode degrees of freedom according to Alice and Bob’s measurement results. Alice and Bob can implement hyper-parallel nonlocal Toffoli gates by performing unitary operations according to Charlie’s measurement result. This protocol has the advantage of having high channel capacity by the hyperentangled state as the quantum channel.

Quantum Algorithm for Variant Maximum Satisfiability

In this paper, we proposed a novel quantum algorithm for the maximum satisfiability problem. Satisfiability (SAT) is to find the set of assignment values of input variables for the given Boolean function that evaluates this function as TRUE or prove that such satisfying values do not exist. For a POS SAT problem, we proposed a novel quantum algorithm for the maximum satisfiability (MAX-SAT), which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with information on how far the given Boolean function is from the SAT satisfaction. We used Grover's algorithm with a new block called quantum counter in the oracle circuit. The proposed circuit can be adapted for various forms of satisfiability expressions and several satisfiability-like problems. Using the quantum counter and mirrors for SAT terms reduces the need for ancilla qubits and realizes a large Toffoli gate that is then not needed. Our circuit reduces the number of ancilla qubits for the terms T of the Boolean function from T of ancilla qubits to ≈⌈log2⁡T⌉+1. We analyzed and compared the quantum cost of the traditional oracle design with our design which gives a low quantum cost.

Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices

Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] can estimate the ground state energy and prepare the ground state of a quantum Hamiltonian with near-optimal query complexities. However, this is based on a block encoding input model of the Hamiltonian, whose implementation is known to require a large resource overhead. We develop a tool called quantum eigenvalue transformation of unitary matrices with real polynomials (QET-U), which uses a controlled Hamiltonian evolution as the input model, a single ancilla qubit and no multi-qubit control operations, and is thus suitable for early fault-tolerant quantum devices. This leads to a simple quantum algorithm that outperforms all previous algorithms with a comparable circuit structure for estimating the ground state energy. For a class of quantum spin Hamiltonians, we propose a new method that exploits certain anti-commutation relations and further removes the need of implementing the controlled Hamiltonian evolution. Coupled with Trotter based approximation of the Hamiltonian evolution, the resulting algorithm can be very suitable for early fault-tolerant quantum devices. We demonstrate the performance of the algorithm using IBM Qiskit for the transverse field Ising model. If we are further allowed to use multi-qubit Toffoli gates, we can then implement amplitude amplification and a new binary amplitude estimation algorithm, which increases the circuit depth but decreases the total query complexity. The resulting algorithm saturates the near-optimal complexity for ground state preparation and energy estimating using a constant number of ancilla qubits (no more than 3).

Quantum reversible circuits for mathrm{GF}(2^{8}) multiplicative inverse

The synthesis of quantum circuits for multiplicative inverse over operatorname{GF}(2^{8}) are discussed in this paper. We first convert the multiplicative inverse operation in operatorname{GF}(2^{8}) to arithmetic operations in the composite field operatorname{GF}((2^{4})^{2}), and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field operatorname{GF}(2^{4}), where the expressions of multiplication calculation in operatorname{GF}(2^{4}) are given directly in operatorname{GF}(2^{4}) and given through being transformed into the composite field operatorname{GF}((2^{2})^{2}). Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over operatorname{GF}(2^{8}) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.

Multiqubit Toffoli Gates and Optimal Geometry with Rydberg Atoms

Due to its potential for implementing a scalable quantum computer, the multiqubit Toffoli gate lies in the heart of quantum information processing. In this paper, we demonstrate a multiqubit blockade gate with atoms arranged in a three-dimensional spheroidal array. The gate performance is greatly improved by the method of optimizing control-qubit distributions on the spherical surface via evolutionary algorithm, which leads to an enhanced asymmetric Rydberg blockade. This spheroidal configuration, not only preserves the dipole blockade energy well between arbitrary control-target pairs, which keeps the asymmetric blockade error at a very low level, but also manifests an unprecedented robustness to the spatial position variations, leading to a negligible position error. Taking account of intrinsic errors and using typical experimental parameters, we numerically show that a c${}_{6}$not Rydberg gate can be created with a fidelity of 0.992, which is only limited by the Rydberg state decays. Our protocol opens up a platform of higher-dimensional atomic arrays for achieving multiqubit neutral-atom quantum computation.

Quantum Fredkin and Toffoli gates on a versatile programmable silicon photonic chip

Quantum logic gates are backbones of quantum information processing (QIP), wherein the typical three-qubit Fredkin and Toffoli gates are essential in quantum computation and communication. So far, the quantum Fredkin gate has only been demonstrated with pre-entangled input states in free-space optics, which limits its usage for independent input photons. Here, we put forward an exquisite scheme and experimentally perform a proof-of-principle demonstration of three-qubit Fredkin and Toffoli gates on a programmable quantum photonic chip. Our scheme can also be used to realize a series of other two-qubit quantum gates. Our work sheds light on the merits of quantum photonic chip in implementing quantum logic gates, and paves the way for advanced quantum chip processors.