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arXiv:2201.03724

Preparation of 3-qubit states

Oscar Mario Perdomo, Nelson Castañeda, Roger Vogeler

Published Jan 11, 2022• 0 citations
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Abstract

We will call a pure qubit state real if all its amplitudes are real numbers. We show that any real 3-qubit state can be prepared using $R_y(\theta)$ gates and at most four controlled-$Z$ gates, and we conjecture that four is optimal. We also present an algorithm -- different from the 2008 algorithm given by Znidaric, Giraud and Georgeot -- that prepares any 3-qubit state using local gates and at most three controlled-$Z$ gates. Videos showing how our method works for two- and three-qubit states can be found at https://youtu.be/LIdYSs-rE-o and https://youtu.be/Kne0Vq7gyzQ

DOI
10.48550/arxiv.2201.03724

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Abstract Let us say that a three-qubit state u 000 |000⟩ + u 001 |001⟩ + ⋯ + u 111 |111⟩ is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | ϕ ⟩, there exist three angles θ 0 , θ 1 and θ 2 such that R y ( θ 2 ) ⊗ R y ( θ 1 ) ⊗ R y ( θ 0 )| ϕ ⟩ is a three-qubit of the form λ 1 |000⟩ + λ 2 |011⟩ + λ 3 |101⟩ + λ 4 |110⟩ + λ 5 |111⟩ with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (Acín et al 2000 Phys. Rev. Lett. 85 1560; Acín et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643).

Journal of Physics A Mathematical and Theoretical•Oct 8, 2021•2 citations
We will call a pure qubit state real if all its amplitudes are real numbers. We show that any real 3-qubit state can be prepared using Ry(θ)R_y(\theta)Ry​(θ) gates and at most four controlled-ZZZ gates, and we conjecture that four is optimal. We also present an algorithm -- different from the 2008 algorithm given by Znidaric, Giraud and Georgeot -- that prepares any 3-qubit state using local gates and at most three controlled-ZZZ gates. Videos showing how our method works for two- and three-qubit states can be found at https://youtu.be/LIdYSs-rE-o and https://youtu.be/Kne0Vq7gyzQ
In this paper, we present a new and algebraically simple algorithm that prepares any pure three-qubit state using only local gates and at most three controlled-[Formula: see text] gates. In 2008, Znidaric et al. already showed that three is the optimal number of controlled-[Formula: see text] gates required for the preparation of three-qubit states if in addition to those gates only local gates are used. If we restrict the local gates to [Formula: see text] gates, our algorithm provides a way to prepare any pure three-qubit state [Formula: see text] with real amplitudes using at most four controlled-[Formula: see text] gates when its hyperdeterminant is negative and three or fewer controlled-[Formula: see text] gates otherwise. We conjecture the existence of three-qubit states with real amplitudes that cannot be prepared using only [Formula: see text] gates and less than four controlled-[Formula: see text] gates.
We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1∣21|21∣2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.
The preparation of nnn-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when mmm ancillary qubits are available. The reduction in complexity is due to the use of a single operator ΛΛΛ for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.
From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as [ϕ⟩=λ1[00⟩+λ2[11⟩[ϕ\rangle=λ_1 [00\rangle+λ_2 [11\rangle[ϕ⟩=λ1​[00⟩+λ2​[11⟩ with λ1λ_1λ1​ and λ2λ_2λ2​ real numbers. For 3-qubits states, it is known, PRL 2000 85, that up to local gates every 3-qubit state can be written as [ϕ⟩=λ1[000⟩+λ2eiθ[100⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[ϕ\rangle=λ_1 [000\rangle+λ_2 e^{i θ}[100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle[ϕ⟩=λ1​[000⟩+λ2​eiθ[100⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ with λi≥0λ_i\ge0λi​≥0 and 0≤θ≤π0\le θ\le π0≤θ≤π. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form [ϕ⟩=λ1[000⟩+λ2[100⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[ϕ\rangle=λ_1 [000\rangle+λ_2 [100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle[ϕ⟩=λ1​[000⟩+λ2​[100⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ by using local gates in the orthogonal group (the group generated by Ry(θ)R_y(θ)Ry​(θ) and XXX gates). We also show that, up to local gates in the orthogonal group, every 3-qubit with real amplitudes can be written as [ϕ⟩=λ1[000⟩+λ2[011⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[ϕ\rangle=λ_1 [000\rangle+λ_2 [011\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle[ϕ⟩=λ1​[000⟩+λ2​[011⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ with the λiλ_iλi​ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}
Abstract Let us say that a three-qubit state u 000 |000⟩ + u 001 |001⟩ + ⋯ + u 111 |111⟩ is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | ϕ ⟩, there exist three angles θ 0 , θ 1 and θ 2 such that R y ( θ 2 ) ⊗ R y ( θ 1 ) ⊗ R y ( θ 0 )| ϕ ⟩ is a three-qubit of the form λ 1 |000⟩ + λ 2 |011⟩ + λ 3 |101⟩ + λ 4 |110⟩ + λ 5 |111⟩ with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (Acín et al 2000 Phys. Rev. Lett. 85 1560; Acín et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643).