QCV2Appendix - 1 Simulation of Quantum Circuits The set of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Simulation of Quantum Circuits The set of unitary operations performed by quantum gates is continuous; thus, the problem of defining a finite, and hopefully small, set of universal quantum gates is slightly more convoluted than the universality of classical gates. In quantum computing we say that a set of quantum gates is universal if any unitary operation can be approximated with ar- bitrary accuracy by a circuit including only gates from this discrete set. We notice that in the case of quantum circuits we have relaxed our requirement of exact simulation. The Solovay- Kitaev theorem tells us that sufficiently good approximations for quantum gates exist. We now discuss the three stages of the protocol to simulate a quantum circuit operating on n qubits using only CNOT, H, S and T gates. Stage 1 . We define a two-level unitary matrix as a transformation involving two or fewer computational basis states. Then we show that any unitary transformation A ∈ H 2 n can be carried out by unitary transformations U 1 ,U 2 ,...U k ,...U m , with m ≤ 2 n- 1, which act only upon two or fewer computational basis states. First, we present an important property of two-level unitary transformations: The matrix representation of a two-level unitary transformation must have at most two non-zero off- diagonal elements. Indeed, the transformation carried out by such a matrix should affect only two projections of a vector. For example, consider two matrices in H 4 , U 1 and U 2 , acting upon an ensemble of two qubits in state | ψ i = α | 00 i + α 1 | 01 i + α 2 | 10 i + α 3 | 11 i : U 1 | ψ i = x y 0 0 z w 0 0 0 0 1 0 0 0 0 1 α α 1 α 2 α 3 = xα + yα 1 zα + wα 1 α 2 α 3 and U 2 | ψ i = x y 0 0 z w 0 0 u 0 1 v 0 0 0 1 α α 1 α 2 α 3 = xα + yα 1 zα + wα 1 uα + α 2 + vα 3 α 3 We notice that in the second case, when there are four non-zero off-diagonal elements three components of the vector are affected by the transformation U 2 . If U is a unitary transformation of one qubit then the elements of the matrix: U = x y z w ¶ , x,y,z,w ∈ C must satisfy the following relations: | x | 2 + | y | 2 = 1 | z | 2 + | w | 2 = 1 xz * + yw * = 0 x * z + y * w = 0 . Indeed, U is unitary thus UU † = U † U = I : 1 x y z w ¶ x * z * y * w * ¶ = 1 0 0 1 ¶ We now examine a two-qubit transformation U ∈ H 4 and consider a decomposition: U = U † 1 U † 2 U † 3 U † 4 U † 5 U † 6 = ⇒ U † = U 6 U 5 U 4 U 3 U 2 U 1 . with U k , 1 ≤ k ≤ 6, given by: U 1 = x y 0 0 z w 0 0 0 0 1 0 0 0 0 1 , U 2 = 1 0 0 0 x y z w 0 0 0 1 , U 3 = 1 0 0 0 0 1 0 0 0 0 x y 0 0 z w , U 4 = x y 0 1 0 0 z w 0 0 0 1 , U 5 = x 0 0 y 0 1 0 0...
View Full Document

This note was uploaded on 08/08/2011 for the course COT 6600 taught by Professor Staff during the Fall '08 term at University of Central Florida.

Page1 / 13

QCV2Appendix - 1 Simulation of Quantum Circuits The set of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online