scholarly journals Causal and compositional structure of unitary transformations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 511
Author(s):  
Robin Lorenz ◽  
Jonathan Barrett
Keyword(s):  
Unitary Transformation ◽  
Quantum Information Processing ◽  
Causal Structure ◽  
Quantum Circuit ◽  
New Formalism ◽  
Compositional Structure ◽  
Unitary Transformations ◽  
Output System ◽  
New Feature ◽  
Circuit Decomposition

The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system A to output system B, system A cannot influence system B. Conversely, given a unitary U with a no-influence relation from input A to output B, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of U with no path from A to B. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident simultaneously. To address this, we introduce a new formalism of `extended circuit diagrams', which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A causally faithful extended circuit decomposition, representing a unitary U, is then one for which there is a path from an input A to an output B if and only if there actually is influence from A to B in U. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary's respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

scholarly journals UNITARY-CIRCUIT SEMANTICS FOR MEASUREMENT-BASED COMPUTATIONS

2010 ◽  
Vol 08 (01n02) ◽  
pp. 1-91 ◽  
Author(s):  
NIEL DE BEAUDRAP
Keyword(s):  
Efficient Algorithm ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
Circuit Model ◽  
Quantum States ◽  
Decision Problems ◽  
Input And Output ◽  
Unitary Transformations ◽  
Quantum Computations

One-way measurement based quantum computations (1WQC) may describe unitary transformations, via a composition of CPTP maps which are not all unitary themselves. This motivates the following decision problems. Is it possible to determine whether a "quantum-to-quantum" 1WQC procedure (having non-trivial input and output subsystems) performs a unitary transformation? Is it possible to describe precisely how such computations transform quantum states, by translation to a quantum circuit of comparable complexity? In this article, we present an efficient algorithm for transforming certain families of measurement-based computations into a reasonable unitary circuit model, in particular without employing the principle of deferred measurement.

Unitary Transformations

1954 ◽  
Vol 6 ◽  
pp. 69-72 ◽  
Author(s):  
B. E. Mitchell
Keyword(s):  
Canonical Form ◽  
Unitary Transformation ◽  
General Pattern ◽  
Jordan Form ◽  
Unitary Matrix ◽  
Complex Matrices ◽  
Triangular Form ◽  
Unitary Transformations

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

Computational model underlying the one-way quantum computer

2002 ◽  
Vol 2 (6) ◽  
pp. 443-486
Author(s):  
R. Raussendorf ◽  
H. Briegel
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Computational Model ◽  
Quantum Logic ◽  
Quantum Information Processing ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Single Step ◽  
Clifford Group ◽  
The One

In this paper we present the computational model underlying the one-way quantum computer which we introduced recently [Phys. Rev. Lett. {\bf{86}}, 5188 (2001)]. The one-way quantum computer has the property that any quantum logic network can be simulated on it. Conversely, not all ways of quantum information processing that are possible with the one-way quantum computer can be understood properly in network model terms. We show that the logical depth is, for certain algorithms, lower than has so far been known for networks. For example, every quantum circuit in the Clifford group can be performed on the one-way quantum computer in a single step.

scholarly journals A Quantum Circuit Design for Grover’s Algorithm

2002 ◽  
Vol 57 (8) ◽  
pp. 701-708 ◽  
Author(s):  
Zijian Diao ◽  
M. Suhail Zubairy ◽  
Goong Chen
Keyword(s):  
Circuit Design ◽  
Cavity Qed ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
The State ◽  
Quantum Phase ◽  
Grover’S Algorithm ◽  
Mathematical Proofs ◽  
Grover's Algorithm

We present a circuit design realizing Grover’s algorithm based on 1-bit unitary gates and 2-bit quantum phase gates implementable with cavity QED techniques. In the first step, we express the circuit block which performs a key unitary transformation that flips only the sign of the state |11 · · · 11〉 using 1-bit and 2-bit gates. The Grover’s iteration operator can then be constructed using this key unitary transformation twice, plus other operations involving only 1-bit unitary gates on each qubit. Mathematical proofs are given to justify that the cricuiting satisfies the desired operator properties.

scholarly journals Towards optimization of quantum circuits

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Michal Sedlák ◽  
Martin Plesch
Keyword(s):  
Quantum Information ◽  
Quantum Information Processing ◽  
Quantum Circuit ◽  
Quantum Gate ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Unitary Operation ◽  
Short Sequence ◽  
Toffoli Gates ◽  
Universal Procedure

AbstractAny unitary operation in quantum information processing can be implemented via a sequence of simpler steps — quantum gates. However, actual implementation of a quantum gate is always imperfect and takes a finite time. Therefore, searching for a short sequence of gates — efficient quantum circuit for a given operation, is an important task. We contribute to this issue by proposing optimization of the well-known universal procedure proposed by Barenco et al. [Phys. Rev. A 52, 3457 (1995)]. We also created a computer program which realizes both Barenco’s decomposition and the proposed optimization. Furthermore, our optimization can be applied to any quantum circuit containing generalized Toffoli gates, including basic quantum gate circuits.

scholarly journals Improved Superconducting Qubit State Readout by Path Interference

2021 ◽  
Vol 38 (11) ◽  
pp. 110303
Author(s):  
Zhiling Wang ◽  
Zenghui Bao ◽  
Yukai Wu ◽  
Yan Li ◽  
Cheng Ma ◽  
...  
Keyword(s):  
Quantum Information Processing ◽  
Quantum Circuit ◽  
Qubit State ◽  
Microwave Cavity ◽  
Single Shot ◽  
Superconducting Qubit ◽  
Constructive Interference ◽  
State Discrimination ◽  
Convenient Approach ◽  
Photon States

High fidelity single shot qubit state readout is essential for many quantum information processing protocols. In superconducting quantum circuit, the qubit state is usually determined by detecting the dispersive frequency shift of a microwave cavity from either transmission or reflection. We demonstrate the use of constructive interference between the transmitted and reflected signal to optimize the qubit state readout, with which we find a better resolved state discrimination and an improved qubit readout fidelity. As a simple and convenient approach, our scheme can be combined with other qubit readout methods based on the discrimination of cavity photon states to further improve the qubit state readout.

scholarly journals Superadiabatic population transfer in a three-level superconducting circuit

2019 ◽  
Vol 5 (2) ◽  
pp. eaau5999 ◽  
Author(s):  
Antti Vepsäläinen ◽  
Sergey Danilin ◽  
Gheorghe Sorin Paraoanu
Keyword(s):  
Excited State ◽  
Ground State ◽  
Microwave Pulse ◽  
Quantum Information Processing ◽  
Quantum Circuit ◽  
Population Transfer ◽  
Adiabatic Passage ◽  
Two Photon ◽  
Superconducting Circuit ◽  
Stimulated Raman Adiabatic Passage

Adiabatic manipulation of the quantum state is an essential tool in modern quantum information processing. Here, we demonstrate the speedup of the adiabatic population transfer in a three-level superconducting transmon circuit by suppressing the spurious nonadiabatic excitations with an additional two-photon microwave pulse. We apply this superadiabatic method to the stimulated Raman adiabatic passage, realizing fast and robust population transfer from the ground state to the second excited state of the quantum circuit.

scholarly journals Householder methods for quantum circuit design

2016 ◽  
Vol 94 (2) ◽  
pp. 150-157 ◽  
Author(s):  
Jesús Urías ◽  
Diego A. Quiñones
Keyword(s):  
Tensor Product ◽  
Circuit Design ◽  
Quantum Circuit ◽  
Quantum Gates ◽  
Single Type ◽  
Unitary Operation ◽  
Gray Codes ◽  
Combined Procedure ◽  
Unitary Transformations ◽  
Assembly Procedure

Algorithms to resolve multiple-qubit unitary transformations into a sequence of simple operations on one-qubit subsystems are central to the methods of quantum-circuit simulators. We adapt Householder’s theorem to the tensor-product character of multi-qubit state vectors and translate it to a combinatorial procedure to assemble cascades of quantum gates that recreate any unitary operation U acting on n-qubit systems. U may be recreated by any cascade from a set of combinatorial options that, in number, are not lesser than super-factorial of 2n, [Formula: see text]. Cascades are assembled with one-qubit controlled-gates of a single type. We complement the assembly procedure with a new algorithm to generate Gray codes that reduce the combinatorial options to cascades with the least number of CNOT gates. The combined procedure —factorization, gate assembling, and Gray ordering — is illustrated on an array of three qubits.

Quantum information processing

2009 ◽  
Author(s):  
Stephen Barnett
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Unitary Transformation ◽  
Quantum Information Processing ◽  
Quantum Computer ◽  
Quantum Evolution ◽  
Quantum Computers ◽  
Digital Electronics ◽  
Logic Operations ◽  
And Control

We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.

scholarly journals DIAMAGNETIC INEQUALITIES FOR SYSTEMS OF NONRELATIVISTIC PARTICLES WITH A QUANTIZED FIELD

1996 ◽  
Vol 08 (02) ◽  
pp. 185-203 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Quantum Electrodynamics ◽  
Unitary Transformation ◽  
Product Formula ◽  
Nelson Model ◽  
Trotter Product Formula ◽  
Gauge Transformations ◽  
Effective Potentials ◽  
Unitary Transformations ◽  
Quantized Field ◽  
Bogoliubov Transformations

By unitary transformations (gauge transformations, the Bogoliubov transformations) and the strong Trotter product formula, diamagnetic inequalities for the Pauli-Fierz model of quantum electrodynamics (QED) and the Nelson model are derived. In the Nelson model, the unitary transformation defines effective potentials. Moreover, the infimum of the spectrum of Hamiltonians for these models are estimated and some generalized Kato’s inequalities are obtained.

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