Elementary Gates Research Articles

OptQC: An optimized parallel quantum compiler

The software package Qcompiler (Chen and Wang 2013) provides a general quantum compilation framework, which maps any given unitary operation into a quantum circuit consisting of a sequential set of elementary quantum gates. In this paper, we present an extended software OptQC, which finds permutation matrices P and Q for a given unitary matrix U such that the number of gates in the quantum circuit of U=QTPTU′PQ is significantly reduced, where U′ is equivalent to U up to a permutation and the quantum circuit implementation of each matrix component is considered separately. We extend further this software package to make use of high-performance computers with a multiprocessor architecture using MPI. We demonstrate its effectiveness in reducing the total number of quantum gates required for various unitary operators. Program summaryProgram title: OptQCCatalogue identifier: AEUA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUA_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 178435No. of bytes in distributed program, including test data, etc.: 491574Distribution format: tar.gzProgramming language: Fortran, MPI.Computer: Any computer with Fortran compiler and MPI library.Operating system: Linux.Classification: 4.15.Nature of problem: It aims to minimize the number of quantum gates required to implement a given unitary operation.Solution method: It utilizes a threshold-based acceptance strategy for simulated annealing to select permutation matrices P and Q for a given unitary matrix U such that the number of gates in the quantum circuit of U=QTPTU′PQ is minimized, where U′ is equivalent to U up to a permutation. The decomposition of a unitary operator is performed by recursively applying the cosine–sine decomposition.Running time: Running time increases with the size of the unitary matrix, as well as the prescribed maximum number of iterations for qubit permutation selection and the subsequent simulated annealing algorithm. Running time estimates are provided for each example in Section 4. All simulation results presented in this paper are obtained from running the program on the Fornax supercomputer managed by iVEC@UWA with Intel Xeon X5650 CPUs.

Efficient quantum simulation of fermionic and bosonic models in trapped ions

We analyze the efficiency of quantum simulations of fermionic and bosonic models in trapped ions. In particular, we study the optimal time of entangling gates and the required number of total elementary gates. Furthermore, we exemplify these estimations in the light of quantum simulations of quantum field theories, condensed-matter physics, and quantum chemistry. Finally, we show that trapped-ion technologies are a suitable platform for implementing quantum simulations involving interacting fermionic and bosonic modes, paving the way for overcoming classical computers in the near future.

DUMMY TITLE FOR ALIASING 10.1186/epjqt9

We analyze the efficiency of quantum simulations of fermionic and bosonic models in trapped ions. In particular, we study the optimal time of entangling gates and the required number of total elementary gates. Furthermore, we exemplify these estimations in the light of quantum simulations of quantum field theories, condensed-matter physics, and quantum chemistry. Finally, we show that trapped-ion technologies are a suitable platform for implementing quantum simulations involving interacting fermionic and bosonic modes, paving the way for overcoming classical computers in the near future.

Design of Logic Gates and Flip-Flops in High-Performance FinFET Technology

With the emergence of nonplanar CMOS devices at the 22-nm node and beyond, it is highly likely that multigate device adoption will occur in a high-performance process technology, owing to the increased performance and area benefits. In this paper, for the first time, we evaluate symmetric (Symm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> ) and asymmetric (Asymm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> ) gate-workfunction FinFETs head to head in a high-performance process, using technology computer-aided design 3-D device simulations. We demonstrate that Asymm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> shorted-gate (a-SG) n/p-FinFETs, which use both workfunctions corresponding to typical high-performance metal-gate n/p-FinFETs, are promising, as they can yield over two orders of magnitude lower leakage without excessive degradation in ON-state current, in comparison to Symm- ΦG shorted-gate (SG) FinFETs, placing them in a better position than back-gate biased independent-gate (IG) FinFETs for leakage reduction. Thereafter, we explore the design space of FinFET logic gates, latches, and flip-flops, for optimal tradeoffs in leakage versus delay and temperature, using mixed-mode 2-D device simulations. Elementary logic gates (such as INV, NAND2, NOR2, XOR2, and XNOR2) using Asymm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> SG-mode FinFETs appear to be located optimally in the leakage-delay spectrum, in comparison to the most versatile configurations possible by mixing corresponding Symm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> SG- and IG-mode FinFETs. Latches and flip-flops, however, require an astute combination of Symm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> and Asymm-Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> FinFETs to optimize leakage, delay, and setup time simultaneously.

Selecting efficient phase estimation with constant-precision phase shift operators

We investigate the cost of three phase estimation procedures that require only constant-precision phase shift operators. The cost is in terms of the number of elementary gates, not just the number of measurements. Faster phase estimation requires the minimal number of measurements with a \log * factor of reduction when the required precision n is large. The arbitrary constant-precision approach (ACPA) requires the minimal number of elementary gates with a minimal factor of 14 of reduction in comparison to Kitaev's approach. The reduction factor increases as the precision gets higher in ACPA. Kitaev's approach is with a reduction factor of 14 in comparison to the faster phase estimation in terms of elementary gate counts.

Quantum computer with atomic logical qubits encoded on macroscopic three-level systems in common quantum electrodynamic cavity

We propose an effective realization of the universal set of elementary quantum gates in solid state quantum computer based on macroscopic (or mesoscopic) resonance systems — multiatomic coherent ensembles, squids or quantum dots in quantum electrodynamic cavity. We exploit an encoding of logical qubits by the pairs of the macroscopic two- or three-level atoms that is working in a Hilbert subspace of all states inherent to these atomic systems. In this subspace, logical single qubit gates are realized by the controlled reversible transfer of single atomic excitation in the pair via the exchange of virtual photons and by the frequency shift of one of the atomic ensembles in a pair. In the case of two-level systems, the logical two-qubit gates are performed by the controlling of Lamb shift magnitude in one atomic ensemble, allowing/blocking the excitation transfer in a pair, respectively, that is controlled by the third atomic system of another pair. When using three-level systems, we describe the NOT-gate in the atomic pair controlled by the transfer of working atomic excitation to the additional third level caused by direct impact of the control pair excitation. Finally, we discuss advantages of the proposed physical system for accelerated computation of some useful quantum gates.

Efficient universal quantum computation with auxiliary Hilbert space

We propose a scheme to construct the efficient universal quantum circuit for qubit systems with the assistance of possibly available auxiliary Hilbert spaces. An elementary two-ququart gate, termed the controlled-double-not gate, is proposed first in ququart (four-level) systems, and its physical implementation is illustrated in the four-dimensional Hilbert spaces built by the path and polarization states of photons. Then an efficient universal quantum circuit for ququart systems is constructed using the gate and the quantum Shannon decomposition method. By introducing auxiliary two-dimensional Hilbert spaces, the universal quantum circuit for qubit systems is finally achieved using the result obtained in ququart systems with the lowest complexity.

Free-surface flow simulations for discharge-based operation of hydraulic structure gates

We combine non-hydrostatic flow simulations of the free surface with a discharge model based on elementary gate flow equations for decision support in the operation of hydraulic structure gates. A water level-based gate control used in most of today's general practice does not take into account the fact that gate operation scenarios producing similar total discharged volumes and similar water levels may have different local flow characteristics. Accurate and timely prediction of local flow conditions around hydraulic gates is important for several aspects of structure management: ecology, scour, flow-induced gate vibrations and waterway navigation. The modelling approach is described and tested for a multi-gate sluice structure regulating discharge from a river to the sea. The number of opened gates is varied and the discharge is stabilized with automated control by varying gate openings. The free-surface model was validated for discharge showing a correlation coefficient of 0.994 compared to experimental data. Additionally, we show the analysis of computational fluid dynamics (CFD) results for evaluating bed stability and gate vibrations.

Complexity Analysis of Quantum Walk Based Search Algorithms

We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O($\sqrt{n}$ log($n$)) elementary 2-qubit quantum gate operations to find a marked node.

Experimental Realization of Nonadiabatic Holonomic Quantum Computation

Because of its geometric nature, holonomic quantum computation is fault tolerant against certain types of control errors. Although proposed more than a decade ago, the experimental realization of holonomic quantum computation is still an open challenge. In this Letter, we report the first experimental demonstration of nonadiabatic holonomic quantum computation in a liquid NMR quantum information processor. Two noncommuting one-qubit holonomic gates, rotations about x and z axes, and the two-qubit holonomic CNOT gate are realized by evolving the work qubits and an ancillary qubit nonadiabatically. The successful realizations of these universal elementary gates in nonadiabatic holonomic quantum computation demonstrates the experimental feasibility of this quantum computing paradigm.

Implementation of Toffoli gates with a single asymmetric HeisenbergXYinteraction

The Toffoli gate, which has practical applications in the implementation of practical quantum algorithms and quantum error correction, is usually constructed from concatenated two-qubit or qubit-qutrit gates. Here we show that the Toffoli gate for three qubits can be implemented through a single $\mathit{XY}$ interaction plus two single-qubit rotations. This method greatly simplifies the experimental implementation of the Toffoli gate and promises a much higher fidelity compared to those based on elementary gate decomposition. We present an example for the implementation of the required $\mathit{XY}$ interaction in cavity QED.

Efficient quantum communication under collective noise

We introduce a new quantum communication protocol for the transmission of quantum information under collective noise. Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented. The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, $m$. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described by an arbitrary discrete group $G$, i.e.~with a discrete number, $\lvert G\rvert$, of possible noise operators, perfect transmission at a rate $m/(m+r)$ is achieved with an overhead that scales at most as $\mathcal{O}(d^r)$ where the number of auxiliary qudits, $r$, depends solely on the group in question. Moreover, this overhead is independent of the number of transmitted qudits, $m$. For certain groups, e.g.~cyclic groups, we find that the overhead scales only linearly with the number of group elements $|G|$.

Minimal and robust composite two-qubit gates with Ising-type interaction

We construct a minimal robust controlled-not gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number ($N=3$) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust swap gate cannot be constructed with $N=3$ but requires $N=6$ elementary two-qubit gates.

Implementing gate operations between uncoupled qubits in linear nearest neighbor arrays using a learning algorithm

We propose a new scheme to implement gate operations in a one dimensional linear nearest neighbor array, by using dynamic learning algorithm. This is accomplished by training quantum system using a back propagation technique, to find the system parameters that implement gate operations directly. The key feature of our scheme is that, we can reduce the computational overhead of a quantum circuit by finding the parameters to implement the desired gate operation directly, without decomposing them into a sequence of elementary gate operations. We show how the training algorithm can be used as a tool for finding the parameters for implementing controlled-NOT (CNOT) and Toffoli gates between next-to-nearest neighbor qubits in an Ising-coupled linear nearest neighbor system. We then show how the scheme can be used to find parameters for realizing swap gates first, between two adjacent qubits and then, between two next-to-nearest-neighbor qubits, in each case without decomposing it into 3 CNOT gates. Finally, we show how the scheme can be extended to systems with non-diagonal interactions. To demonstrate, we train a quantum system with Heisenberg interactions to find the parameters to realize a swap operation.

A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

Synthesis of multivalued quantum logic circuits by elementary gates

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 Shannon decomposition (QSD), the most powerful technique for the synthesis of generic qubit circuits, to the qudit case. A comparison of ququart (d=4) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.

Information coding of exciting laser pulses in an optical echo-processor

We report the possibility of controlling the distribution of quantum bits within an inhomogeneously broadened line of a resonant transition in recording and transforming information in optical echo-processors. We consider the efficiency of realisation of the elementary logic XOR gate based on a two-pulse excitation of a resonant medium with phase memory. The encoded information is incorporated into the temporal shape of laser pulses in the form of amplitude modulation of an ‘echelon’ of present (‘1’) and absent (‘0’) pulse-codes for obtaining more efficient logic elements that reduce the noise in a quantum communication channel.

Quantum binary field inversion: improved circuit depth via choice of basis representation

Finite fields of the form ${\mathbb F}_{2^m}$ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth $\bigO(m\log(m))$. To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields ${\mathbb F}_{2^m}$ based on an elementary gate set that is useful for fault-tolerant implementation.

Generation of SWAP gate between two remote cavities via an optical fiber by adiabatic passage

In this paper, we have looked at the system of two cavities, each of them having N atoms, and both connected by an optical fiber. The process for the construction of a SWAP gate among two spatially separated cavities and N atoms via a stimulated Raman adiabatic passage has been presented. It does not require a composition of elementary gates from a universal set and it shows that a highly reliable SWAP gate is achievable based on adiabatic passage. The SWAP operation involves three types of atomic states: with 0 excitation, single excitation and two excitations. The two-excitation state corresponds to double excitation of the atoms. Possible pathways for information transfer for this double excitation the explained. In this scheme the last state involves 1 atom in the excited state and N - 1 atoms in the ground state of cavity one and one atom in the excited state and N - 1 atoms in the ground state of cavity 2 and this is not connected to the rest of the 7 states.

Simplified construction and physical realization ofn-qubit controlled phase gates

We show that with the assistance of a third level of qubits an $n$-qubit phase gate can be constructed from $2n\ensuremath{-}4$ two-qutrit conditional swap gates, a single qutrit-qubit controlled phase gate, and two single-qutrit operations. Unlike previous schemes, our scheme uses the auxiliary level to ``expose'' some state to the qutrit-qubit controlled phase gate, instead of using it to ``hide'' states from the conditional dynamics. Neither the number of the additional levels nor that of single-qutrit operations needs to increase with $n$. We propose a physical implementation of the required elementary gates in cavity QED and show that the total gate time may be greatly reduced compared with that required in previous methods.