Factoring larger integers with fewer qubits via quantum annealing with optimized parameters

RSA cryptography is based on the difficulty of factoring large integers, which is an NP-hard(and hence intractable) problem for a classical computer. However, Shor's algorithm shows that its complexity is polynomial for a quantum computer, although technical difficulties mean that practical quantum computers that can tackle integer factorizations of meaningful size are still a long way away. Recently, Jiang et al. proposed a transformation that maps the integer factorization problem onto the quadratic unconstrained binary optimization(QUBO) model. They tested their algorithm on a D-Wave 2000 Q quantum annealing machine, raising the record for a quantum factorized integer to 376289 with only 94 qubits. In this study, we optimize the problem Hamiltonian to reduce the number of qubits involved in the final Hamiltonian while maintaining the QUBO coefficients in a reasonable range, enabling the improved algorithm to factorize larger integers with fewer qubits. Tests of our improved algorithm using D-Wave's hybrid quantum/classical simulator qbsolv confirmed that performance was improved, and we were able to factorize 1005973, a new record for quantum factorized integers, with only 89 qubits. In addition, our improved algorithm can tolerate more errors than the original one. Factoring 1005973 using Shor's algorithm would require about 41 universal qubits,which current universal quantum computers cannot reach with acceptable accuracy. In theory, the latest IBM Q System OneTM(Jan. 2019) can only factor up to 10-bit integers, while the D-Wave have a thousand-fold advantage on the factoring scale. This shows that quantum annealing machines, such as those by D-Wave, may be close to cracking practical RSA codes, while universal quantum-circuit-based computers may be many years away from attacking RSA.     

Implementing logic gates and the Deutsch-Jozsa quantum algorithm by two-dimensional NMR using spin- and transition-selective pulses

Quantum logical operations using two-dimensional NMR have recently been described using the scalar coupling evolution technique [J. Chem. Phys. 109, 10603 (1998)]. In the present paper, we describe the implementation of quantum logical operations using two-dimensional NMR, with the help of spin- and transition-selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are also implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. The Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been implemented using spin- and transition-selective pulses after creating a coherent superposition state in the two-dimensional methodology.     

Experimental implementation of a fixed-point duality quantum search algorithm in the nuclear magnetic resonance quantum system

In this work, we demonstrated a fixed-point quantum search algorithm in the nuclear magnetic resonance (NMR) system. We constructed the pulse sequences for the pivotal operations in the quantum search protocol. The experimental results agree well with the theoretical predictions. The generalization of the scheme to the arbitrary number of qubits has also been given.     

A quantum circuit design of AES requiring fewer quantum qubits and gate operations

Advanced Encryption Standard (AES) is one of the most widely used block ciphers nowadays, and has been established as an encryption standard in 2001. Here we design AES-128 and the sample-AES (S-AES) quantum circuits for deciphering. In the quantum circuit of AES-128, we perform an affine transformation for the SubBytes part to solve the problem that the initial state of the output qubits in SubBytes is not the |0>^⊗8 state. After that, we are able to encode the new round sub-key on the qubits encoding the previous round sub-key, and this improvement reduces the number of qubits used by 224 compared with Langenberg et al.’s implementation. For S-AES, a complete quantum circuit is presented with only 48 qubits, which is already within the reach of existing noisy intermediate-scale quantum computers.     

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q=2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen-Shannon divergence, the relative entropy of coherence, and the l₁ norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.     

High-fidelity quantum operations on superconducting qubits in the presence of noise

We present a scheme for implementing quantum operations with superconducting qubits. Our approach uses a "coupler" qubit to mediate a controllable interaction between data qubits, pulse sequences which strongly mitigate the effects of 1/f flux noise, and a high-Q resonator-based local memory. We develop a Monte Carlo simulation technique capable of describing arbitrary noise-induced dephasing and decay, and demonstrate in this system a set of universal gate operations with O(10(-5)) error probabilities in the presence of experimentally measured levels of 1/f noise. We then add relaxation and quantify the decay times required to maintain this error level.     

Transferring quantum entangled states between multiple single-photon-state qubits and coherent-state qubits in circuit QED

We present a way to transfer maximally- or partially-entangled states of n single-photon-state (SPS) qubits onto ncoherent-state (CS) qubits, by employing 2nmicrowave cavities coupled to a superconducting flux qutrit. The two logic states of a SPS qubit here are represented by the vacuum state and the single-photon state of a cavity, while the two logic states of a CS qubit are encoded with two coherent states of a cavity. Because of using only one superconducting qutrit as the coupler, the circuit architecture is significantly simplified. The operation time for the state transfer does not increase with the increasing of the number of qubits. When the dissipation of the system is negligible, the quantum state can be transferred in a deterministic way since no measurement is required. Furthermore, the higher-energy intermediate level of the coupler qutrit is not excited during the entire operation and thus decoherence from the qutrit is greatly suppressed. As a specific example, we numerically demonstrate that the high-fidelity transfer of a Bell state of two SPS qubits onto two CS qubits is achievable within the present-day circuit QED technology. Finally, it is worthy to note that when the dissipation is negligible, entangled states of n CS qubits can be transferred back onto n SPS qubits by performing reverse operations. This proposal is quite general and can be extended to accomplish the same task, by employing a natural or artificial atom to couple 2nmicrowave or optical cavities.     

Universal quantum control based on parametric modulation in superconducting circuits

As superconducting quantum circuits are scaling up rapidly towards the noisy intermediate-scale quantum(NISQ)era, the demand for electronic control equipment has increased significantly. To fully control a quantum chip of N qubits,the common method based on up-conversion technology costs at least 2 × N digital-to-analog converters(DACs) and N IQ mixers. The expenses and complicate mixer calibration have become a hinderance for intermediate-scale quantum control.Here we propose a universal control scheme for superconducting circuits, fully based on parametric modulation. To control N qubits on a chip, our scheme only requires N DACs and no IQ mixer, which significantly reduces the expenses. One key idea in the control scheme is to introduce a global pump signal for single-qubit gates. We theoretically explain how the universal gates are constructed using parametric modulation. The fidelity analysis shows that parametric single-qubit(two-qubit) gates in the proposed scheme can achieve low error rates of 10~(4), with a gate time of about 60 ns(100 ns).     

Universal quantum logic gates in a scalable Ising spin quantum computer

We consider the model of quantum computer, which is represented as a Ising spin lattice, where qubits (spin-half systems) are separated by the isolators (two spin-half systems). In the idle mode or at the single bit operations the total spin of isolators is 0. There are no need of complicated protocols for correcting the phase and probability errors due to permanent interaction between the qubits. We present protocols for implementation of universal quantum gates with the rectangular radio-frequency pulses.     

The hybrid quantum computer

In this paper, we look at the possibility of realizing one-way quantum computing through a hybrid quantum computing architecture based on stationary qubits inside an optical cavity and flying qubits (photons). It has been shown that direct qubit-qubit interactions for two-qubit gate implementations can be replaced by the experimentally less demanding generation of single photons on demand and linear optics photon pair measurements. The outcomes of these measurements indicate either the completion of the gate or the presence of the original qubits, such that the operation can be repeated until success.     

Iterative quantum algorithm for distributed clock synchronization

Clock synchronization is a well-studied problem with many practical and scientific applications.We propose an arbitrary accuracy iterative quantum algorithm for distributed clock synchronization using only three qubits.The n bits of the time difference between two spatially separated clocks can be deterministically extracted by communicating only O(n) messages and executing the quantum iteration process n times based on the classical feedback and measurement operations.Finally,we also give the algorithm using only two qubits and discuss the success probability of the algorithm.     

Nanoscale superconducting quantum bits

Various physical systems were proposed for quantum information processing. Among those nanoscale devices appear most promising for integration in electronic circuits and large-scale applications. We discuss Josephson junction circuits in two regimes where they can be used for quantum computing. These systems combine intrinsic coherence of the superconducting state with control possibilities of single-charge circuits. In the regime where the typical charging energy dominates over the Josephson coupling, the low-temperature dynamics is limited to two states differing by a Cooper-pair charge on a superconducting island. In the opposite regime of prevailing Josephson energy, the phase (or flux) degree of freedom can be used to store and process quantum information. Under suitable conditions the system reduces to two states with different flux configurations. Several qubits can be joined together into a register. The quantum state of a qubit register can be manipulated by voltage and magnetic field pulses. The qubits are inevitably coupled to the environment. However, estimates of the phase coherence time show that many elementary quantum logic operations can be performed before the phase coherence is lost. In addition to manipulations, the final state of the qubits has to be read out. This quantum measurement process can be accomplished using a single-electron transistor for charge Josephson qubits, and a d.c.-SQUID for flux qubits. Recent successful experiments with superconducting qubits demonstrate for the first time quantum coherence in macroscopic systems.     

Universal quantum computation using all-optical hybrid encoding

By employing displacement operations, single-photon subtractions, and weak cross-Kerr nonlinearity, we propose an alternative way of implementing several universal quantum logical gates for all-optical hybrid qubits encoded in both single-photon polarization state and coherent state. Since these schemes can be straightforwardly implemented only using local operations without teleportation procedure, therefore, less physical resources and simpler operations are required than the existing schemes. With the help of displacement operations, a large phase shift of the coherent state can be obtained via currently available tiny cross-Kerr nonlinearity. Thus, all of these schemes are nearly deterministic and feasible under current technology conditions, which makes them suitable for large-scale quantum computing.     

Physical implementation of three-qubit gates on a separate quantum particle

A virtual spin formalism is suggested to demonstrate that a single quantum particle possessing eight suitable discrete energy levels can be used for storing three information qubits and organizing on them a universal set of logical operations that are necessary for constructing an arbitrary quantum algorithm. The formalism can be practically implemented on a nuclear spin 7/2 subjected to resonance rf pulses. A single-pulse realization is found for all quantum gates of a universal set, including a three-qubit gate.     

Transfer of quantum entangled states between superconducting qubits and microwave field qubits

Transferring entangled states between matter qubits and microwave-field (or optical-field) qubits is of fundamental interest in quantum mechanics and necessary in hybrid quantum information processing and quantum communication. We here propose a way for transferring entangled states between superconducting qubits (matter qubits) and microwave-field qubits. This proposal is realized by a system consisting of multiple superconducting qutrits and microwave cavities. Here, „qutrit” refers to a three-level quantum system with the two lowest levels encoding a qubit while the third level acting as an auxiliary state. In contrast, the microwave-field qubits are encoded with coherent states of microwave cavities. Because the third energy level of each qutrit is not populated during the operation, decoherence from the higher energy levels is greatly suppressed. The entangled states can be deterministically transferred because measurement on the states is not needed. The operation time is independent of the number of superconducting qubits or microwave-field qubits. In addition, the architecture of the circuit system is quite simple because only a coupler qutrit and an auxiliary cavity are required. As an example, our numerical simulations show that high-fidelity transfer of entangled states from two superconducting qubits to two microwave-field qubits is feasible with present circuit QED technology. This proposal is quite general and can be extended to transfer entangled states between other matter qubits (e.g., atoms, quantum dots, and NV centers) and microwave- or optical-field qubits encoded with coherent states.     

Fault tolerant quantum computation with nondeterministic gates

In certain approaches to quantum computing the operations between qubits are nondeterministic and likely to fail. For example, a distributed quantum processor would achieve scalability by networking together many small components; operations between components should be assumed to be failure prone. In the ultimate limit of this architecture each component contains only one qubit. Here we derive thresholds for fault-tolerant quantum computation under this extreme paradigm. We find that computation is supported for remarkably high failure rates (exceeding 90%) providing that failures are heralded; meanwhile the rate of unknown errors should not exceed 2 in 10(4) operations.     

Enhanced quantum state detection efficiency through quantum information processing

We investigate theoretically and experimentally how quantum state-detection efficiency is improved by the use of quantum information processing (QIP). Experimentally, we encode the state of one 9Be(+) ion qubit with one additional ancilla qubit. By measuring both qubits, we reduce the state-detection error in the presence of noise. The deviation from the theoretically allowed reduction is due to infidelities of the QIP operations. Applying this general scheme to more ancilla qubits suggests that error in the individual qubit measurements need not be a limit to scalable quantum computation.     

A hybrid quantum encoding algorithm of vector quantization for image compression

Many classical encoding algorithms of vector quantization (VQ) of image compression that can obtain global optimal solution have computational complexity O(N). A pure quantum VQ encoding algorithm with probability of success near 100% has been proposed, that performs operations 45\sqrt{N} times approximately. In this paper, a hybrid quantum VQ encoding algorithm between the classical method and the quantum algorithm is presented. The number of its operations is less than \sqrt{N} for most images, and it is more efficient than the pure quantum algorithm.     

A proposal for the realization of universal quantum gates via superconducting qubits inside a cavity

A family of quantum logic gates is proposed via superconducting (SC) qubits coupled to a SC-cavity. The Hamiltonian for SC-charge qubits inside a single mode cavity is considered. Three- and two-qubit operations are generated by applying a classical magnetic field with the flux. Therefore, a number of quantum logic gates are realized. Numerical simulations and calculation of the fidelity are used to prove the success of these operations for these gates.     

Quantum simulation of τ-anti-pseudo-Hermitian two-level systems

Different from the Hermitian case, non-Hermitian (NH) systems have novel properties and strongly relate to open and dissipative quantum systems. In this work, we investigate how to simulate τ-anti-pseudo-Hermitian systems in a Hermitian quantum device using linear combinations of unitaries and duality quantum algorithm. Specifying the τ to time-reversal (T) and parity-time-reversal (PT) operators, we construct the two NH two-level systems, design quantum circuits including three qubits, and decide the quantum gates explicitly in detail. We also calculate the success probabilities of the simulation. Experimental implementation can be expected in small quantum simulator.