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Single-Pulse Preparation of the Uniform Superpositional State used in Quantum Algorithms

Berman, G P, Borgonovi, F, Izrailev, F M, Tsifrinovich, V I

Published Jun 21, 2000• 3 citations
OpenAlex

Abstract

We examine a single-pulse preparation of the uniform superpositional wave function, which includes all basis states, in a spin quantum computer. The effective energy spectrum and the errors generated by this pulse are studied in detail. We show that, in spite of the finite width of the energy spectrum bands, amplitude and phase errors can be made reasonably small.

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We examine a single-pulse preparation of the uniform superpositional wave function, which includes all basis states, in a spin quantum computer. The effective energy spectrum and the errors generated by this pulse are studied in detail. We show that, in spite of the finite width of the energy spectrum bands, amplitude and phase errors can be made reasonably small.
Most quantum algorithms assume some specific initial state in superposition of basis states before performing the desired application-specific computations. The preparation of such states itself requires a computation performed by a quantum circuit. In this paper, we investigate the automatic state preparation of a specific subset of quantum states that are uniform superpositions over a subset of basis states, called uniform quantum states. We exploit that such states can be represented using Boolean functions and present a recursive algorithm based on functional decomposition. When using binary decision diagrams as function representation, we can enable fast and scalable quantum state preparation with respect to the size of the decision diagram. We show that the algorithm can find quantum circuits for functions, where state-of-the-art algorithms cannot be applied anymore.
The quantum search algorithm can be looked at as a technique for synthesizing a particular kind of superposition-one whose amplitude is concentrated in a single basis state. This basis state is defined by a binary function f(&xmacr;) that is nonzero in this desired basis state and zero everywhere else. This paper extends the quantum search algorithm to an algorithm that can create an arbitrarily specified superposition on a space of size N in O(sqrt[N] ) steps. The superposition is specified by a complex valued function f(&xmacr;) that specifies the desired amplitude of the system in basis state &xmacr;.
We present a procedure for quantum teleportation in a nuclear spin quantum computer in which quantum logic gates are implemented by using selective electromagnetic pulses. A sequence of pulses is combined with single-spin measurements in the ${\ensuremath{\sigma}}^{z}$ basis for fast transfer of information in a spin quantum computer. We simulated this procedure for quantum teleportation in a nuclear spin chain with a large number (201) of spins. The systematic errors generated in the process of teleportation due to the non-resonant effects are analyzed in detail. We demonstrate that a $``2\ensuremath{\pi}k''$ method provides a significant reduction of errors.
A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.
Near-term quantum devices provide only finite-shot measurements and prepare imperfect, contaminated states. This motivates algorithms that convert samples into reliable low-energy estimates without full tomography or exhaustive measurements. We propose Active Sampling Sample-based Quantum Diagonalization (AS-SQD), framing SQD as an active learning problem: given measured bitstrings, which additional basis states should be included to efficiently recover the ground-state energy? SQD restricts the Hamiltonian to a selected set of basis states and classically diagonalizes the restricted matrix. However, naive SQD using only sampled states suffers from bias under finite-shot sampling and excited-state contamination, while blind random expansion is inefficient as system size grows. We introduce a perturbation-theoretic acquisition function based on Epstein--Nesbet second-order energy corrections to rank candidate basis states connected to the current subspace. At each iteration, AS-SQD diagonalizes the restricted Hamiltonian, generates connected candidates, and adds the most valuable ones according to this score. We evaluate AS-SQD on disordered Heisenberg and Transverse-Field Ising (TFIM) spin chains up to 16 qubits under a preparation model mixing 80\% ground state and 20\% first excited state. Furthermore, we validate its robustness against real-world state preparation and measurement (SPAM) errors using physical samples from an IBM Quantum processor. Across simulated and hardware evaluations, AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion. Detailed ablation studies demonstrate that physics-guided basis acquisition effectively concentrates computation on energetically relevant directions, bypassing exponential combinatorial bottlenecks.
It has been shown that, starting from the state |0>, in the general case, an arbitrary quantum state |ψ> cannot be prepared with exponential precision in polynomial time. However, we show that for the important special case when |ψ> represents discrete values of some real, continuous function ψ(x), efficient preparation is possible by applying the eigenvalue estimation algorithm to a Hamiltonian which has ψ(x) as an eigenstate. We construct the required Hamiltonian explicitly and present an iterative algorithm for removing unwanted superpositions from the output state in order to reach |ψ> within exponential accuracy. The method works under very general conditions and can be used to provide the quantum simulation algorithm with very accurate and general starting states.