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Chinese Science Bulletin

Restricted allowable generalized quantum gates

Cao Huai

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

An allowable generalized quantum gate(introduced by Long,Liu and Wang)has the form of U 1 0 d kk k c U -= ∑,where Uk's are unitary operators on a Hilbert space H and 1 0 1 d kkc -= |∑|≤and 1 k |c|≤(0≤k≤d-1).In this work we consider a kind of AGQGs,called restricted allowable generalized quantum gates(RAGQGs),satisfying 1 0 0 1 d kk c -= ∑||≤.Some properties of the set RAGQG(H)of all RAGQGs on H are established.Especially,we prove that the extreme points of RAGQG(H)are exactly unitary operators on H and that B(H)=R+RAGQG(H).

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An allowable generalized quantum gate(introduced by Long,Liu and Wang)has the form of U 1 0 d kk k c U -= ∑,where Uk's are unitary operators on a Hilbert space H and 1 0 1 d kkc -= |∑|≤and 1 k |c|≤(0≤k≤d-1).In this work we consider a kind of AGQGs,called restricted allowable generalized quantum gates(RAGQGs),satisfying 1 0 0 1 d kk c -= ∑||≤.Some properties of the set RAGQG(H)of all RAGQGs on H are established.Especially,we prove that the extreme points of RAGQG(H)are exactly unitary operators on H and that B(H)=R+RAGQG(H).
In this paper, we give a characterization of generalized quantum gates.We also show that many important operators are generalized quantumgates, moreover, some of these operators can be represented as theconvex combination of only two unitary operators. Our results answerwhat kinds of operations a duality quantum computing admits. Wepoint out that the set of all generalized quantum gates coincides withthe set of all restricted allowable generalized quantum gates. Thus,our results are also valid for restricted allowable generalized quantumgates
Quantum computing with qudits, quantum systems with d>2d > 2d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels ddd, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary ddd, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
For numerous applications of quantum theory it is desirable to be able to apply arbitrary unitary operations on a given quantum system. However, in particular situations only a subset of unitary operations is easily accessible. This raises the question of what additional unitary gates should be added to a given gate set in order to attain physical universality, i.e., to be able to perform arbitrary unitary transformation on the relevant Hilbert space. In this work, we study this problem for three paradigmatic cases of naturally occurring restricted gate sets: (A) particle-number preserving bosonic linear optics, (B) particle-number preserving fermionic linear optics, and (C) general (not necessarily particle-number preserving) fermionic linear optics. Using tools from group theory and control theory, we classify, in each of these scenarios, what sets of gates are generated, if an additional gate is added to the set of allowed transformations. This allows us to solve the universality problem completely for arbitrary number of particles and for arbitrary dimensions of the single-particle Hilbert space.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of 4n4^{n}4n-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study universality for quantum computations by quantum operations on mixed states.
We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere S1(H)S_1(\mathcal H)S1​(H) of a Hilbert space H\mathcal HH . For a measure on S1(H)S_1(\mathcal H)S1​(H) , we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital C∗\mathrm C^*C∗ -algebra are characterized in terms of extreme points. We also study extreme points extr(E1)\mathrm{extr}(\mathcal E^1)extr(E1) of the unit ball E1\mathcal E^1E1 of a normed ideal operator space ⟨E,∥⋅∥E⟩\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle⟨E,∥⋅∥E​⟩ on H\mathcal HH . If U∈extr(E1)U\in\mathrm{extr}(\mathcal E^1)U∈extr(E1) for some unitary operator U∈B(H)U\in\mathcal{B}(\mathcal H)U∈B(H) , then V∈extr(E1)V\in\mathrm{extr}(\mathcal E^1)V∈extr(E1) for all unitary operators V∈B(H)V\in\mathcal{B}(\mathcal H)V∈B(H) . In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.
In this paper, we give the most general duality gates, or generalized quantum gates in duality quantum computers. Here we show by explicit construction that a n-bit duality quantum computer with d slits can be simulated perfectly with an ordinary quantum computer with n qubits and one auxiliary qudit. Using this model, we give the most general form of duality gates which is of the form Σ d- 1 i = 0 p i U i , and the p i 's are complex numbers with module less or equal to 1 and constrained by |Σ i p i |≤1.