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arXiv:quant-ph/9908052 · Physical Review A

Implementing unitary operators in quantum computation

Jae‐Hyun Kim, Jae‐Seung Lee, Soonchil Lee

Published Feb 16, 2000• 46 citations
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Abstract

We present a general method which expresses a unitary operator by the product of operators allowed by the Hamiltonian of spin-1/2 systems. In this method, the generator of an operator is found first, and then the generator is expanded by the base operators of the product operator formalism. Finally, the base operators disallowed by the Hamiltonian, including more than two-body interaction operators, are replaced by allowed ones by the axes transformation and coupling order reduction technique. This method directly provides pulse sequences for the nuclear magnetic resonance quantum computer, and can be generally applied to other systems.

DOI
10.1103/physreva.61.032312

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We present a general method which expresses a unitary operator by the product of operators allowed by the Hamiltonian of spin-1/2 systems. In this method, the generator of an operator is found first, and then the generator is expanded by the base operators of the product operator formalism. Finally, the base operators disallowed by the Hamiltonian, including more than two-body interaction operators, are replaced by allowed ones by the axes transformation and coupling order reduction technique. This method directly provides pulse sequences for the nuclear magnetic resonance quantum computer, and can be generally applied to other systems.
Under mild conditions the experimentally accessible Hamiltonians in ensembles of N coupled, nondegenerate spins-½ are shown to generate the entire Lie group SU(2 N ). This is the case e.g., in nuclear magnetic resonance (NMR) spectroscopy [1], where quantum computing gates [2] are implemented by unitary propagators, viz the ‘rf-pulse sequences’. The maximum achievable coherence transfer amplitude under unitary Hamiltonian dynamics of ensembles translates into a minimum Euclidean distance: the minimal distance between the unitary orbit of some given initial state represented by a density operator (or its signal-relevant components collected in a matrix A) and a given final state or observable (or its components C †). This distance relates to the C-numerical range of A, W C(A) := {tr(UAU -1 C)|U ∈ SU(2 N )}, and its largest absolute value, the C-numerical radius of A, rC(A):=maxU∣tr{UAU−1C}∣[3] r_C \left( A \right) : = \mathop {{\rm{max}}}\limits_U |tr\left\{ {U AU^{ - 1} C} \right\}|\left[ {\rm{3}} \right] rC​(A):=Umax​∣tr{UAU−1C}∣[3] [3]. The geometry of the C-numerical range includes as limiting cases the notions of ensembles comprising entangled as well as classically mixed components. In the latter, entropy and relative entropy relate to Euclidean distances. Given two arbitrary matrices A,C ∈ Mat n (C), the first gradient-flow-based computer algorithm [4-6] for minimising the Euclidean distance between the unitary orbit of A and C † (or rather C) in the general case is presented.
The improved quantum scheduling algorithm proposed by Grover has been generalized using the generalized quantum search algorithm, in which a unitary operator replaces the Walsh-Hadamard transform, and π/2π/2π/2 phase rotations replace the selective inversions, in order to make the quantum scheduling algorithm suitable for more cases. Our scheme is realized on a nuclear magnetic resonance (NMR) quantum computer. Experimental results show a good agreement between theory and experiment.
We introduce a scheme for realizing arbitrary controlled-unitary operations in a two-qubit system. If the $2\ifmmode\times\else\texttimes\fi{}2$ unitary matrix is special unitary (has unit determinant), the controlled-unitary gate operation can be realized in a single pulse operation. The pulse in our scheme will constitute varying one of the parameters of the system between an arbitrarily maximum and a ``calculated'' minimum value. This parameter will constitute the variable parameter of the system while the other parameters, which include the coupling between the two qubits, will be treated as fixed parameters. The values of the parameters are what we solve for using our approach in order to realize an arbitrary controlled-unitary operation. We further show that the computational complexity of the operation is no greater than that required for a controlled-NOT (CNOT) gate. Since conventional schemes realize a controlled-unitary operation by breaking it into a sequence of single-qubit and CNOT gate operations, our method is an improvement because we not only require less time duration, but also fewer control lines to implement the same operation. To demonstrate improvement over other schemes, we show, as examples, how two controlled-unitary operations, one being the controlled-Hadamard gate, can be realized in a single pulse operation using our scheme. Furthermore, our method can be applied to a wide range of coupling schemes and can be used to realize gate operations between two qubits coupled via Ising, Heisenberg, and anisotropic interactions.
Unitary transformations play a fundamental role in many-body physics, and except for special cases, they are not expressible in closed form. We present closed-form expressions for unitary transformations generated by a single fermionic operator for Hermitian and anti-Hermitian generators. We demonstrate the usefulness of these expressions in formal analyses of unitary transformations and numerical applications to Hamiltonian downfolding in quantum computing and Heisenberg dynamics. This work paves the way for new analytical treatments of unitary transformations and numerical many-body methods for fermions.
Experiments in coherent nuclear and electron magnetic resonance and quantum computing in general correspond to control of quantum-mechanical systems, guiding them from initial to final target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and decoherence and to optimize the sensitivity of the experiments. Here, we derive a time-optimal sequences as fundamental building blocks to synthesize unitary transformations. Such sequences can be widely implemented on various physical systems, including the simulation of effective Hamiltonians for topological quantum computing on spin lattices. Experimental demonstrations are provided for a system consisting of three nuclear spins.