Solution 4 for USTC class Physics of Quantum Information

Transcription

1 Solution 4 for USTC class Physics of Quantum Information Shuai Zhao, Xin-Yu Xu, Kai Chen National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, , P.R. China 1. Please write down the difference between quantum error correction and classical error correction. Read page 3-6 in the lecture QIP2018chapt 5 1 Kai Chen.pdf for reference. 2. (1) Please write down the qunatum error-correction condition. (2) Consider the three qubit bit filp code, with corresponding projector P = The noise process that this code protects against has operation elements { (1 p) 3 I, p(1 p) 2 X 1, p(1 p) 2 X 2, p(1 p) 2 X 3 }, where p is the probability that one bit filps. Note that this quantum operation is not trace-preserving, since we have omitted operation elements corresponding to bit flips on two and three qubits. Verify the quantum error-correction conditions for this code and noise process. (1) Read page in the lecture QIP2018chapt 5 1 Kai Chen.pdf for reference. (2) The operation elements of the noise process is {E 0, E 1, E 2, E 3 }, where E 0 = (1 p) 3 I and E i = p(1 p) 2 X i for i = 1, 2, 3. If i j, P X i X jp = 0, if i = j, P X i X jp = P 2 = P. So P E i E jp = α ij P, in which α 00 = (1 p) 3, α ii = p(1 p) 5 for i = 1, 2, 3 and α ij = 0(i j). α is an Hermitian matrix, so the three qubit bit filp code and noise process satisfies the quantum error-correction conditions.

2 2 3. (1) For 4-qubit GHZ state ψ = ( ) / 2, please write down its linearly independent stabilizers. (2) For 4-qubit cluster state ψ = ( )/2, please write down its linearly independent stabilizers. (1) An example is as following: g 1 = ZZII, g 2 = IZZI, g 3 = IIZZ, g 4 = XXXX. (Other answers are welcome.) (2) g 1 = XZII, g 2 = ZXZI, g 3 = IZXZ, g 4 = IIZX. 4. Please draw the quantum circuit of the 3-qubit bit flip code, and certify that it can encode the qubit a 0 + b 1 to a b 111. The quantum circuit is as following: The encoding progress is as following: ψ 0 0 = (a 0 + b 1 ) 0 0 C-NOT GGGGGGGGGGA(a 00 + b 11 ) 0 C-NOT GGGGGGGGGGA(a b 111 ) (1) 5. For 9-qubit Shor code, its logical bit code is 0 L = ( )( )( )/2 2,

3 3 1 L = ( )( )( )/2 2. (1) Please give all the generators of the stabilizers; (2) Please draw the encoding quantum circuit; (3) For a bit/phase flip error of a certain bit, how to detect and correct it? Please take the bit flip error and phase flip error for example, write down the program of error detection and correction. (1) The generators of the stabilizers are as following: (2) The encoding quantum circuit is as following:

4 4 (3) Please read page 433 of Nielsen s Quantum Computation and Quantum Information, or page 80 of the Chinese version translated by Qian-Chuan Zhao. 6. Please write down a group of universial quantum logical gate set. A group of universial quantum logical gate set is as following: T(45-degree rotation of Z), H(Hadamard gate), C-NOT. 7. Please write down the DiVincenzo criterion that quantum computer implementation must satisfy. (1) Scalability: A scalable physical system with well characterized parts, usually qubits. (2) Initialization: The ability to initialize the system in a simple fiducial state. (3) Control: The ability to control the state of the computer using sequences of elementary universal gates. (4) Stability: Decoherence times much longer than gate times, together with the

5 ability to suppress decoherence through error correction and fault-tolerant computation. (5) Measurement:The ability to read out the state of the computer in a convenient product basis. 8. Please write down the matrix form of the C-NOT gate, and the state resulting from the application of this gate to four Bell states. The matrix form of the C-NOT gate is as following: Denote the four Bell states as β xy = 0, y + ( 1)x 1, ȳ 2 5, then C-NOT( β 00 ) = ( ) 0 / 2 C-NOT( β 01 ) = ( ) 1 / 2 C-NOT( β 10 ) = ( 0 1 ) 0 / 2 C-NOT( β 11 ) = ( 0 1 ) 1 / 2 9. Please construct the quantum SWAP gate to swap two qubits using the C-NOT gate. The circuit swapping two qubits is as following: To see that this circuit accomplishes the swap operation, note that the sequence of gates has the following sequence of effects on a computational basis state a, b,

6 6 where all additions are done modulo Please design a quantum circuit which converts the state 00, 01, 10, 11 into four Bell states. The quantum circuit to create Bell state is as following: The proof is omitted. 11. Please design a quantum circuit to perform full Bell state measurement, i.e. to distinguish four Bell states by projective measurement at 0, 1 basis. The quantum circuit to perform full Bell state measurement is as following: The proof is omitted. 12. Please draw the quantum circuit of Deutsch algorithm, and analysis how it works. Please read page of lecture QIP2018chapt 6 1 Kai Chen.pdf for reference.