Authors Faisal Shah Khan Marek A. Perkowski Slides prepared by Faisal Shah Khan

ه A qudit replaces a classical dit as an information unit in d-valued quantum computing. ه A qudit represented as a unit vector in the state space, which is a complex projective d-dimensional Hilbert space, ه In the computational basis, the basis vectors of are written in Dirac notation as where 1 is in the i-th position Overview

ه An arbitrary vector in can be expressed as a linear combination ه The real number is the probability that the state vector will be in i-th basis state upon measurement. ه When the state spaces of n qudits of different d-valued dimensions are combined via their algebraic tensor product, the result is a n qudit hybrid state space where is the state space of the qudit. Overview

ه The computational basis for H will consist of all possible tensor products of the computational basis vectors of the component state spaces. ه If all the different are assigned the same value d, the resulting state space is that of n d-valued qudits. ه The evolution of state space changes the state of the qudits via the action of a unitary (length preserving) operator on the qudits. ه A unitary operator can be represented by a square unitary evolution matrix. ه For the hybrid state space H, an evolution matrix will have size while the evolution matrix for will be of size Overview

ه In the context of quantum logic synthesis, an evolution matrix is a quantum logic circuit that needs to be realized by a universal set of quantum logic gates. ه It is well established (Brylinski, Muthukrishnan) that sets of one and two qudit quantum gates are universal. Hence, the synthesis of an evolution matrix requires that the matrix be decomposed to the level of unitary matrices acting on one or two qudits. ه Unitary matrix decomposition methods like the QR factorization and the Cosine Sine decomposition (CSD) from matrix perturbation theory have been used for 2- valued (binary) and 3-valued (Ternary) quantum logic synthesis. ه Mottonen et. al, Shende et. al – Binary CSD sythesis ه Khan and Perkowski – Ternary CSD synthesis Overview

Cosine-Sine Decomposition (CSD)

n-qubit (Binary) Quantum Logic Synthesis via CSD ه In this case, unitary W matrices are of size. ه Let so that ه Now the CSD gives W decomposed as ه Each block in the block matrices of the CSD of W are of size

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) ه Note that the CSD can be iteratively applied to the unitary block diagonals that occur in the decomposition at each stage. ه The iteration stops when the block are of size 2 x 2. ه However, there may be local optimizations involving a CNOT (4x4 matrix) and 2 x 2 gates.

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) ه Shende et al and Mottonen et al give the following realization of the factors in the CSD at each iterative level: a) Block diagonal matrices are Quantum Multiplexers. b) The cosine-sine matrices are uniformly controlled rotations, a variation of the multiplexer.

n-qubit Quantum Multiplexer

Uniformly (n-1)-controlled rotation

n-qubit Quantum Multiplexer in Dirac / Matrix notation where is the i-th qubit in he circuit, and both block matrices are of size ه Depending on whether, (M-1) reduces to (M-1)

(n-1)-controlled rotation in Dirac / Matrix notation (R-1)

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) M U V U V A 2-qubit quantum multiplexer R0R1 A 2-qubit uniformly 1-controlled rotation 0 1

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) Example 1. A 2-qubit quantum multiplexer matrix. The first qubit controls the second. ه If the first qubit is 0, then U is applied to the second qubit. ه If the first qubit is 1, then V is applied to the second qubit. ه For n-qubits, a quantum multiplexer will control the lowest (n-1) qubits via the top qubit.

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) Example 2. A 2-qubit uniformly controlled rotation matrix. This 4 x 4 matrix acts on the tensor product of the two qubits

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) Calculations give: Let Then (1) becomes (1) Let Then (1) becomes

n-qubit (Binary) Quantum Logic Synthesis via CSD (continued) ه A uniformly controlled rotation is essentially a multiplexer. ه In our example, the bottom qubit controls the top. ه For n-qubits, a uniformly controlled rotations is a multiplexer in which the lowest qubit controls the top (n-1) qubits.

n-qutrit (Ternary) Quantum Logic Synthesis via CSD ه In this case, unitary matrices W are of size ه Let so that ه Now the CSD gives W decomposed as ه The top corner blocks in the diagonal matrices are of size ه The lower corner blocks, are of size Both C and S matrices are of size (2)

n-qutrit (Ternary) Quantum Logic Synthesis via CSD (continued) Uniformly controlled rotation around z-axis in Uniformly controlled rotation around x-axis in Multiplexer

n-qutrit Quantum Multiplexer

(n-1)-controlled R_x rotation

ه Consider the matrix d-valued Quantum Logic Synthesis vis CSD