Discussion

0.7606 −0.6493

T

, and its state fidelity is 66.31%.

Figure 4.6: The result of the HHL algorithm on the real device. The left diagram shows the state fidelity of each stage and the right diagram shows the measurement result of the answer |xi.

123

The error on a quantum simulator is exactly 0%, so the error comes from the imperfect quantum device. Note that the initialization and the phase estimation let the state fidelity decrease slowly, but the reciprocal rotation doesn’t. The error of a two-qubit controlled gate is ten times larger than a single qubit gate. The reciprocal rotation stage in my circuit includes many controlled rotations, this is the reason why the state fidelity decrease rapidly in that stage.

4.5 Discussion

123

Base on the properties of quantum computers, I have presented the whole pro-cess of the quantum algorithm for solving a linear system problem by quantum computers. Then, I have implemented them successfully on a quantum simulator and a real quantum computer. In fact, there are some problems that need to be overcome before the realization of quantum linear system algorithm to be useful for general cases. First, the algorithm I proposed for preparing an arbitrary unitary gate is general but may not be the most efficient. This optimization problem is stills a research topic nowadays. Once the unitary gate W in the WZP algorithm can be prepared in O(n³) time for both classical and quantum cases, rather than

O(n⁴), the whole quantum process is better than the classical algorithm. Second, the final answer to the quantum linear system algorithms are stored in the ampli-tude of quantum state, but there is no efficient way to read out the ampliampli-tudes for many qubits so far. If we can design an algorithm that stores the answer in the basis instead of amplitudes, then the readout step can be efficient. Another way to avoid the problem is to view QLSA’s as a subroutine of another quantum algorithm, so we don’t need to read out the amplitudes. Third, the errors in a real quantum computer come from initializations, quantum gates, and measurements. The ini-tialization and measurement errors depend on the width of a quantum circuit (the number of qubits), and the quantum gate errors depend on the depth (the number of gates in series). Besides the improvement of devices, we can trade off between the qubits used and the gates used. Another method is to do the quantum error cor-rection. But this can not be a solution in the present time since the phyical qubits used for logical qubits must be much more than 20 qubits. That is, the quantum error correction scheme can be realized only when the quantum computer has been scaled up to many qubits system.

Chapter 5 Conclusion

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To summarize, the quantum parallelism due to the superposition property and the exponential growth of the Hilbert space due to the tensor product makes tum computers powerful. Further, solving a linear system problem using the quan-tum algorithm has exponential improvement on quanquan-tum computer as long as the previous problems mentioned in Discussion section can be solved, and they can be then applied to the problems in machine learning. Furthermore, I have proposed the algorithm to map a n × 1 vector ~v into quantum states |vi in O(n) time, and the algorithm to map a n × n unitary matrix U into quantum circuit with universal gates in O(n³) time for both classical and quantum cases. The HHL algorithm takes O(s²log(n)) and is efficient for only dealing with a sparse matrix, but the WZP algorithm takes O(polylog(n)) and is efficient for general case. Finally, I use only six qubits to solve a particular 2×2 quantum linear system problem and reach more than 99% fidelity on quantum simulator and 66% on real quantum computer. A key point to achieve this is that the reciprocal rotation circuit I designed is an exact transformation, which can be generalized to serve as an important component for solving an quantum linear system problem with arbitrary n × n dimenional matrix.

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Appendix A

Proof of the matrix factorizations

123

Given a matrix A ∈ R^m×n. The singular value decomposition (SVD) is A = U ΣV^H, where U is an m × m unitary matrix, V is an n × n unitary matrix, and Σ is a rectangular diagonal matrix with decreased sorted singular values σ_i. In this section, we showed the derivation of SVD. Start from the equation

A = U ΣV^H

Hence,

The singular value σ_i and the singular vector v_i are in the equation

A^HA~v_i = σ_i²v~_i. (A.4)

The other part of the answer, singular vector u_i is in the equation

A~v_i = σ_iu~_i, qed. (A.6)

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The next is the vectorization decomposition. Given a matrix A ∈ R^m×n. A = M^HN ||A||_F is called vectorization decomposition (VD). The matrix M is a mn × m isometry matrix, and N is a mn × n isometry matrix. The scalar ||A||_F is the Frobenius norm of A. The i^th column of M is

M ~e_i = ~e_i⊗ A~i H

|| ~A_i|| = ~e_i⊗ 1

|| ~A_i||[Σⁿ_j=1A_ije~_j^H]^H = 1

|| ~A_i||Σⁿ_j=1A_ij^∗~e_i⊗ ~e_j, (A.7)

and the j^th column of N is

N ~ej = a~_F

||A||_F ⊗ ~ej = 1

||A||_FΣ^m_i=1|| ~Ai||~ei⊗ ~ej. (A.8)

The elements of M^HN is i^th row of M^H times j^th column of N , which is equivalent to i^th column of M times j^th column of N . It also becomes

(M^HN )ij = (M ~ei)^∗(N ~ej) = A_ij

||A||_F, qed. (A.9)

Appendix B

Proof of the eigenvalues of W

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Recall the vectorization decomposition A = M^HN ||A||_F and the orthogonal projection generate an unitary operator

W = (2M M^H − I)(2N N^H − I). (B.1)

Since W is a series of reflection, some of its eigenvectors are relate to M ~u_i and N ~v_i, which are called non-trivial eigenvectors ~v^±. First, express the eigenvector ~w_i as the linear combination of the bases spanning the column spaces R(M ) and R(N ).

~

w_i = aM ~u_i+ bN ~v_i, (B.2)

where a and b are two constants. Next, multiply W on the bases. Calculate them by some algebra, then

W N ~v_i = 2bcosθ_i

2M ~u_i− bN ~v_i (B.3)

and

W M ~u_i = ( 4σ_i²

||A||²_F − 1)M ~u_i− 2σ_i

||A||_FN ~v_i. (B.4)

Thus, multiply W on its eigenvector. The result becomes,

W ~wi = [a( 4σ²_i

||A||²_F − 1) + b 2σ_i

||A||_F]M ~ui+ [a−2σ_i

||A||_F − b]N ~vi = λi(aM ~ui+ bN ~vi) (B.5)

Finally, the eigenvalue of W can be derived by the ratio of the constants a and b.

a

b = −e^±iθi/2, λ_i = −2a bcosθ_i

2 − 1 = e^±iθi, qed. (B.6)

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