Circuit Model Of Quantum Computation Research Articles

Construction of two-qubit gates using B Gate

Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and iSWAP gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of B gate, an unexplored non-perfect entangler of the form, exp i π ( σ ˆ x ⨂ σ ˆ x ) 8 + i π ( σ ˆ y ⨂ σ ˆ y ) 16 , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of B gate. Using this condition, we show that the gates that can be generated by two and three applications of B gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of B gate can generate both SWAP and SWAP † which is not possible for CNOT and iSWAP gates; further, we conjecture that three applications of B gate can generate all perfect entanglers. Finally, we discuss about the construction of a three independent parameter universal two-qubit quantum circuit using four B gates that can generate all two-qubit gates. In the end, we mention about the schemes to implement B gate in ion-trap quantum computers.

Addressable Quantum Gates

We extend the circuit model of quantum computation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders and making their geometrical layout explicit: we express the quantum switch and the polarizing beam-splitter within the model. In this context, our main contribution is a full characterization of the anonymity constraints. Indeed, the names used as addresses should not matter beyond the wiring they describe; i.e., quantum evolutions should commute with “renamings.” We show that these quantum evolutions can still act non-trivially upon the names. We specify the structure of “nameblind” matrices.

Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation.

Simulating the unitary dynamics of a quantum system is a fundamental problem of quantum mechanics, in which quantum computers are believed to have significant advantage over their classical counterparts. One prominent such instance is the simulation of electronic dynamics, which plays an essential role in chemical reactions, non-equilibrium dynamics, and material design. These systems are time-dependent, which requires that the corresponding simulation algorithm can be successfully concatenated with itself over different time intervals to reproduce the overall coherent quantum dynamics of the system. In this paper, we quantify such simulation algorithms by the property of being fully-coherent: the algorithm succeeds with arbitrarily high success probability 1 - δ while only requiring a single copy of the initial state. We subsequently develop fully-coherent simulation algorithms based on quantum signal processing (QSP), including a novel algorithm that circumvents the use of amplitude amplification while also achieving a query complexity additive in time t, ln(1/δ), and ln(1/ϵ) for error tolerance ϵ: Θ‖H‖|t|+ln(1/ϵ)+ln(1/δ). Furthermore, we numerically analyze these algorithms by applying them to the simulation of the spin dynamics of the Heisenberg model and the correlated electronic dynamics of an H2 molecule. Since any electronic Hamiltonian can be mapped to a spin Hamiltonian, our algorithm can efficiently simulate time-dependent ab initio electronic dynamics in the circuit model of quantum computation. Accordingly, it is also our hope that the present work serves as a bridge between QSP-based quantum algorithms and chemical dynamics, stimulating a cross-fertilization between these exciting fields.

Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states

While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme, where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N-times Controlled-Z gate illustrates a novel trade-off between space and time complexity.

Exploring the Implementation of Steganography Protocols on Quantum Audio Signals

Two quantum audio steganography (QAS) protocols are proposed, each of which manipulates or modifies the least significant qubit (LSQb) of the host quantum audio signal that is encoded as an FRQA (flexible representation of quantum audio) audio content. The first protocol (i.e. the conventional LSQb QAS protocol or simply the cLSQ stego protocol) is built on the exchanges between qubits encoding the quantum audio message and the LSQb of the amplitude information in the host quantum audio samples. In the second protocol, the embedding procedure to realize it implants information from a quantum audio message deep into the constraint-imposed most significant qubit (MSQb) of the host quantum audio samples, we refer to it as the pseudo MSQb QAS protocol or simply the pMSQ stego protocol. The cLSQ stego protocol is designed to guarantee high imperceptibility between the host quantum audio and its stego version, whereas the pMSQ stego protocol ensures that the resulting stego quantum audio signal is better immune to illicit tampering and copyright violations (a.k.a. robustness). Built on the circuit model of quantum computation, the circuit networks to execute the embedding and extraction algorithms of both QAS protocols are determined and simulation-based experiments are conducted to demonstrate their implementation. Outcomes attest that both protocols offer promising trade-offs in terms of imperceptibility and robustness.

Sequent Calculus Representations for Quantum Circuits

When considering a sequent-style proof system for quantum programs, there are certain elements of quantum mechanics that we may wish to capture, such as phase, dynamics of unitary transformations, and measurement probabilities. Traditional quantum logics which focus primarily on the abstract orthomodular lattice theory and structures of Hilbert spaces have not satisfactorily captured some of these elements. We can start from 'scratch' in an attempt to conceptually characterize the types of proof rules which should be in a system that represents elements necessary for quantum algorithms. This present work attempts to do this from the perspective of the quantum circuit model of quantum computation. A sequent calculus based on single quantum circuits is suggested, and its ability to incorporate important conceptual and dynamic aspects of quantum computing is discussed. In particular, preserving the representation of phase helps illustrate the role of interference as a resource in quantum computation. Interference also provides an intuitive basis for a non-monotonic calculus.

State complexity and quantum computation

The complexity of a quantum state may be closely related to the usefulness of the state for quantum computation. We discuss this link using the tree size of a multiqubit state, a complexity measure that has two noticeable (and, so far, unique) features: it is in principle computable, and non‐trivial lower bounds can be obtained, hence identifying truly complex states. In this paper, we first review the definition of tree size, together with known results on the most complex three and four qubits states. Moving to the multiqubit case, we revisit a mathematical theorem for proving a lower bound on tree size that scales superpolynomially in the number of qubits. Next, states with superpolynomial tree size, the Immanant states, the Deutsch‐Jozsa states, the Shor's states and the subgroup states, are described. We showed that the universal resource state for measurement based quantum computation, the 2D cluster state, has superpolynomial tree size. Moreover, we showed how the complexity of subgroup states and the 2D cluster state can be verified efficiently. The question of how tree size is related to the speed up achieved in quantum computation is also addressed. We show that superpolynomial tree size of the resource state is essential for measurement based quantum computation. The necessary role of large tree size in the circuit model of quantum computation is still a conjecture; and we prove a weaker version of the conjecture. image

On the Circuit Model of Global Adiabatic Search Algorithm

Local adiabatic search has been implemented on the quantum circuit model, and the time steps necessary to guarantee the approximation precision are of the same order as the time complexity of the algorithm. However, it is easy to verify that this conclusion doesn’t hold for global adiabatic search. But we know that adiabatic computing is equivalent to the circuit model of quantum computation. Therefore, there must exist a corresponding quantum circuit that correctly implements such a global adiabatic search. In this Letter, we show that this indeed could be true.

Measurement-based quantum computation cannot avoid byproducts

In the circuit model of quantum computation, a desired unitary gate can be implemented deterministically, whereas in the measurement-based model the unitary gate is implemented up to a byproduct operator. In order to compensate byproducts, following measurement angles must be adjusted, or classical results must be corrected. Such a feed-forwarding requires some classical processing and tuning of the measurement device, which cause the delay of computation and the additional decoherence. We show that if we respect the no-signaling principle, which is one of the most fundamental principles in physics, byproducts cannot be avoided in measurement-based quantum computation. Furthermore, we also show by using the idea of the quantum error correcting code that due to the no-signaling principle, not all byproducts are allowed in measurement-based quantum computation.

Quantum game simulator, using the circuit model of quantum computation

We present a general two-player quantum game simulator that can simulate any two-player quantum game described by a 2 × 2 payoff matrix (two strategy games).The user can determine the payoff matrices for both players, their strategies and the amount of entanglement between their initial strategies. The outputs of the simulator are the expected payoffs of each player as a function of the other player's strategy parameters and the amount of entanglement. The simulator also produces contour plots that divide the strategy spaces of the game in regions in which players can get larger payoffs if they choose to use a quantum strategy against any classical one. We also apply the simulator to two well-known quantum games, the Battle of Sexes and the Chicken game. Program summary Program title: Quantum Game Simulator (QGS) Catalogue identifier: AEED_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEED_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3416 No. of bytes in distributed program, including test data, etc.: 583 553 Distribution format: tar.gz Programming language: Matlab R2008a (C) Computer: Any computer that can sufficiently run Matlab R2008a Operating system: Any system that can sufficiently run Matlab R2008a Classification: 4.15 Nature of problem: Simulation of two player quantum games described by a payoff matrix. Solution method: The program calculates the matrices that comprise the Eisert setup for quantum games based on the quantum circuit model. There are 5 parameters that can be altered. We define 3 of them as constant. We play the quantum game for all possible values for the other 2 parameters and store the results in a matrix. Unusual features: The software provides an easy way of simulating any two-player quantum games. Running time: Approximately 0.4 sec (Region Feature) and 0.3 sec (Payoff Feature) on a Intel Core 2 Duo GHz with 2 GB of memory under Windows XP.

The complexity of quantum spin systems on a two-dimensional square lattice

The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA. In this paper we show that this important problem remains QMA-complete when the interactions of the 2-local Hamiltonian are between qubits on a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate k-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation. Our perturbation method also shows how any stabilizer space associated with a k-local stabilizer (for constant k) can be generated as an approximate ground-space of a 2-local Hamiltonian.

Quantum computer simulator based on the circuit model of quantum computation

A quantum computer simulator is presented. This simulator is an engineering work and no deep understanding of quantum mechanics is required from the user. The simulator is based on the circuit model of quantum computation in which quantum gates act on quantum registers which comprise a number of quantum bits (qubits). The inputs to the simulator are the initial states of the qubits that form a quantum register and the quantum gates applied at each computation step. The inputs are entered through a graphical user interface. The outputs of the simulator are the matrices that represent the quantum register state at each quantum computation step and graphical outputs that show the probability of measuring each one of the possible quantum register base states and the phase of each state at each computation step. The well-known Deutsch's algorithm and the quantum Fourier transform, which is the base of many quantum algorithms, are presented using this simulator. Furthermore, the generation and variation of entanglement during quantum computations can be calculated using this simulator. The quantum computer simulator is a useful tool for the study of quantum computer circuits, quantum computing, and the development of new quantum algorithms.

Entanglement dynamics in one-dimensional quantum cellular automata

Several proposed schemes for the physical realization of a quantum computer consist of qubits arranged in a cellular array. In the quantum circuit model of quantum computation, an often complex series of two-qubit gate operations is required between arbitrarily distant pairs of lattice qubits. An alternative model of quantum computation based on quantum cellular automata (QCA) requires only homogeneous local interactions that can be implemented in parallel. This would be a huge simplification in an actual experiment. We find some minimal physical requirements for the construction of unitary QCA in a 1 dimensional Ising spin chain and demonstrate optimal pulse sequences for information transport and entanglement distribution. We also introduce the theory of non-unitary QCA and show by example that non-unitary rules can generate environment assisted entanglement.