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Almost-classical quantum computers

Michiel Boes, Alexis De Vos, Jan De Beule

Published Jan 1, 2010• 1 citation
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Abstract

By means of a subgroup of the 2 X 2 unitary matrices, i.e. a subgroup Q of U(2), acting on a single qubit, we create a group X, acting on w qubits. If Q equals the group of order 2 consisting of the follower and the inverter, we recover S_{2^w}, i.e. the permutation matrices describing a classical reversible computer acting on w bits. If Q is another group of two 2 X 2 matrices, then a new kind of computing appears.

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By means of a subgroup of the 2 X 2 unitary matrices, i.e. a subgroup Q of U(2), acting on a single qubit, we create a group X, acting on w qubits. If Q equals the group of order 2 consisting of the follower and the inverter, we recover S_{2^w}, i.e. the permutation matrices describing a classical reversible computer acting on w bits. If Q is another group of two 2 X 2 matrices, then a new kind of computing appears.
Classical reversible circuits, acting on www~bits, are represented by permutation matrices of size 2w×2w2^w \times 2^w2w×2w. Those matrices form the group P(2w2^w2w), isomorphic to the symmetric group {\bf S}2w_{2^w}2w​. The permutation group P(nnn), isomorphic to {\bf S}n_nn​, contains cycles with length~ppp, ranging from~1 to L(n)L(n)L(n), where L(n)L(n)L(n) is the so-called Landau function. By Lagrange interpolation between the ppp~matrices of the cycle, we step from a finite cyclic group of order~ppp to a 1-dimensional Lie group, subgroup of the unitary group U(nnn). As U(2w2^w2w) is the group of all possible quantum circuits, acting on www~qubits, such interpolation is a natural way to step from classical computation to quantum computation.
Classical reversible computers on w bits are isomorphic to the (finite) symmetric group S 2 w ; quantum computers on w qubits are isomorphic to the (Lie) unitary group U(2 w ). We investigate and classify groups X which represent computers intermediate between classical reversible computers and quantum computers. Such intermediate groups X may exist in three flavours: finite groups of order larger than (2 w )!, infinite but discrete groups, and Lie groups of dimension smaller than (2 w ). For our purpose, we start from 1-qubit transformations, represented by 2 × 2 unitary matrices. We call this group the creator. Its members are called gates and act on one qubit. Controlled gates are quantum circuits acting on w qubits, such that the 1-qubit transformation (applied to a particular qubit) depends on the state of the w − 1 other qubits. The controlled gates generate the group X of 2 w × 2 w matrices, called the creation. We discuss all creators of order up to 8. Additionally a creator of order 16 and one of order 192 are discussed.
As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit ϕ( θ ) and the 1-qubit NEGATOR circuit N ( θ ). Both are roots of the IDENTITY circuit. Indeed: both ϕ(0) and N (0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2 w ); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2 w ). Here, ZU( n ) and XU( n ) are subgroups of the unitary group U( n ): the group XU( n ) consists of all n × n unitary matrices with all 2 n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU( n ) consists of all n × n unitary diagonal matrices with first entry equal to 1. Any U( n ) matrix can be decomposed into four parts: U = exp( iα ) Z 1 XZ 2 , where both Z 1 and Z 2 are ZU( n ) matrices and X is an XU( n ) matrix. We give an algorithm to find the decomposition. For n = 2 w it leads to a four-block synthesis of an arbitrary quantum computer.
Quantum computation on w qubits is represented by the infinite unitary group U(2 w ); classical reversible computation on w bits is represented by the finite symmetric group S 2 w . In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU( n ) and ZU( n ) of the unitary group U( n ). The former consists of all unitary n × n matrices with all line sums equal to 1; the latter consists of all unitary diagonal n × n matrices with first entry equal to 1. Such a group structure also reveals the relationship between matrix calculus and diagrammatic zx-calculus of quantum circuits.