Unitary gate preparation

3.2.1 Lemma

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Given a unitary matrix U , how to design a quantum circuit using just single-qubit and two-single-qubit gates to performs such operation or transformation? Before solving the problem, the following lemmas are useful.

Lemma 1: Unitary matrix condition

Consider a matrix U. U is unitary ⇔ U’s column vectors are orthonormal

⇔ U’s row vector are orthonormal

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This have been proved for example, in a linear algebra textbook [4]. It is an useful lemma because the statements are both sufficient and necessary. So one can determine whether a matrix is unitary at the first look by this lemma.

Lemma 2: Row exchange elementary matrix

Suppose there is an identity matrix I ∈ C^n×n. Then exchange the i^th row and j^th row of I become a new matrix Eij, the so-called row exchange elementary matrix.

Then, E_ijA is a matrix that exchange i^th row and j^th row of A, where A ∈ C^n×p.

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This is also proved in [4], which is a part of foundation of linear equations theory.

Lemma 3: Any n × n unitary matrix can be decompose into a multiplication chain with ⁿ⁽ⁿ⁻¹⁾₂ elements, where each element has exactly 2 non-trivial bases.

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The lemma has been mentioned in [12]. Here I take a 3 × 3 unitary matrix

U =

as an example. The goal is to express the matrix as a multiplication chain of

3(3−1)

2 = 3 elements. i.e. U = U₁^HU₂^HU₃^H or U3U2U1U = I, where Ui’s are unitary matrix with 2-non trivial bases for all i = 1, 2, 3. First, let

U₁ =

Note that we have eliminated the (2, 1) element. Next, let

U₂ =

so

In this step, we eliminate the (3,1) element. Furthermore, the first column has only one element. So the element (1,1) must be 1 and other elements in the first row must be 0. This can be easily derived by lemma 1. Finally,

U₃ = blkdiag

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Now we analyze the complexity. It takes O(n) time to construct U_i, and the matrix multiplication costs O(n) time since there are only 2n elements that need to be updated. The length of the multiplication chain is O(n²). Thus, the total complexity of decomposed n × n unitary matrix on a classical computer is O(n³).

Definition: Level-2 matrix

A is level-2 matrix if it satisfies: (i) A is 2^c× 2^c unitary matrix, where c is a positive integer, (ii) A is constructed by two non-trivial bases and other standard bases, and (iii) the index difference of the two non-trivial basis is δ = 2^d, where d is an integer.

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For example,

is unitary and has 4 × 4 = 2² × 2² dimensions. There are two non-trivial bases, which are located at the first and the third row(column). Furthermore, the index difference δ = 3 − 1 = 2¹. Hence, it is a level-2 matrix.

Lemma 4: Gray code determines needed level-2 matrix

To decompose a n × n unitary gate, we need n-1 level-2 matrices which are generated by g_log(n). Express g_log(n) in decimal representation and then add 1 to get a vector V. The level-2 matrices has non-trivial bases at the i^th and (i + 1)^th element in V,

∀i = 1, 2, · · · , n − 1, and they are collected into a set called S.

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For example, to decompose a 4 × 4 unitary matrix, consider the Gray code g_log(4) = g₂ =(00,01,11,11), and V=(1,2,4,3). Therefore, the level-2 matrices we need are in the set S = {G^1,2₂ , G^2,4₂ , G^4,3₂ }, and please see Fig 3.3 for their matrix representation. The symbol ”X” in the figure indicates the places of non-trivial bases. Note that the difference between adjacent elements in Gray code is only one bit, so δ is always an integer power of 2, and G^i,j₂ must be a level-2 matrix.

Figure 3.3: The required two-qubit gates for preparing any 4 × 4 unitary operation.

Lemma 5: The quantum circuit of any level-2 matrix is a multi-control gate

Consider g^i,j_log(n) a level-2 matrix, and δ = |i − j|. The target bit locates at [log(δ) + 1]^th qubit (count from down to up), and the type of controller (white or black) is determined by the place of the non-trivial basis regularly.

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In fact, the quantum gate of any n × n level-2 matrix can be constructed easily at the first look, see Fig 3.3. A n-bit multi-control gate also can be decomposed into O(2^log(n)) = O(n) universal gates mentioned in the previous section.

3.2.2 Universal gate decomposition algorithm

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Base on the lemmas, we can decompose any 2^c×2^cunitary matrix into universal gates by the python pseudo code shown in Fig. 3.4.

Figure 3.4: The unitary decomposition algorithm. The input of the algorithm is an unitary matrix and the output is the series of multi-controlled gates

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In the beginning, I test whether U is unitary, and has true dimension. If it isn’t, then return false value; otherwise, then after a series of operations, return a queue containing the elements of the multiplication chain.

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The j^th iteration in the ”for loop” is to clear and reset all of the elements in the j^th row and j^th column to 0 except for U (j, j) which should be 1. In the for loop, check whether U (j, j) is 1, if it is, go to next iteration. Otherwise, there are two cases. The first case is U (j, j) = 0, then we must find a non-zero element in the j^th column and exchange the i^th row and the j^th row by multiplying the row exchange elementary matrix E_ij. So we need to enqueue E_ij. Note that, the order of finding non-zero elements must use the Gray code order because we know only how to implement level-2 matrix. The second case is U (j, j) 6= 1, then we have to clear and reset all of the elements in the j^th row and j^th column to 0 except for U (j, j), just as in lemma 3. But the different thing is that the order of elimination must be the Gray code order too.

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The decomposition on classical computer takes O(n³) time as in lemma 3. The quantum complexity is O(n²) × O(2^log(n)) = O(n³). Although it may not be the most efficient method, but it is a general algorithm that can implement any unitary operation by one-qubit and two-qubit universal quantum gates.

在文檔中 實現任意聯立方程式之量子演算法及其在機器學習上之應用 (頁 39-44)