Householder factorizations of unitary matrices

Urı́as, Jesús

doi:10.1063/1.3451111

Cited by 17 publications

(24 citation statements)

References 11 publications

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“…This becomes a coefficient to the product V j P j (X, I) . B H can be considered as a Householder transformation: An L dimensional Householder matrix requires O(L) number of quantum gates [25,26]. An example U H for a general 8 × 8 Hamiltonian is presented in Fig.2, where P j s are simplified into two CNOT gates.…”

Section: (26)mentioning

confidence: 99%

A generalized circuit for the Hamiltonian dynamics through the truncated series

Daskin

Kais

2018

Quantum Inf Process

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In this paper, we present a method for the Hamiltonian simulation in the context of eigenvalue estimation problems which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, H(t), through the truncated Taylor series method described by Berry et al. [1]. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable t in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires O(Ln) number of quantum gates for the simulation of a molecular Hamiltonian. Here, n is the number of states of a spin orbital, and L is the number of terms in the molecular Hamiltonian and generally bounded by O(n 4 ). We also discuss how to use the circuit in adaptive processes and eigenvalue related problems along with a slight modified version of the iterative phase estimation algorithm. In addition, a simple divide and conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by O(poly(n)) for a matrix with poly(n)−sparsity.

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Section: (26)mentioning

confidence: 99%

A generalized circuit for the Hamiltonian dynamics through the truncated series

Daskin

Kais

2018

Quantum Inf Process

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“…We know that a transformation U is unitary if, and only if, it is realizable as a succession of Householder reflections [7]. The convenience of reflections when conceiving a physical implementation of U , e.g.…”

Section: Time Evolution As a Succession Of Reflectionsmentioning

confidence: 99%

“…For instance, u 0 = 011 does not appear in any of the lists in (13). The choices (12) and (13) determine one of the many possible sequences of reflections in (7) and constitutes, in the next Section, a template for the cascade of quantum gates.…”

Section: Time Evolution As a Succession Of Reflectionsmentioning

confidence: 99%

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Householder methods for quantum circuit design

Urı́as

Quiñones

2016

Can. J. Phys.

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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.

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“…all the way down to U (2). Notable examples are based on beam-splitter transformations [2] or the Householder decompositions [3,4,5]. Although these decompositions can be realized physically by means of multi-beam splitters or Mach-Zehnder interferometers [2], they are not in natural accordance with a multi-qubit architecture.…”

Section: Introductionmentioning

confidence: 99%

Block-ZXZsynthesis of an arbitrary quantum circuit

Vos

Baerdemacker

2016

Phys. Rev. A

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Given an arbitrary 2 w × 2 w unitary matrix U , a powerful matrix decomposition can be applied, leading to four different syntheses of a w-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by H. Führ and Z. Rzeszotnik [Linear Algebra Its Appl. 484, 86 (2015)] generalizing the scaling of single-bit unitary gates (w = 1) to gates with arbitrary value of w. The synthesized circuit consists of controlled one-qubit gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled NOT gates in the case that U is a permutation matrix.

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Householder factorizations of unitary matrices

Cited by 17 publications

References 11 publications

A generalized circuit for the Hamiltonian dynamics through the truncated series

A generalized circuit for the Hamiltonian dynamics through the truncated series

Householder methods for quantum circuit design

Block-ZXZsynthesis of an arbitrary quantum circuit

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