1. Qubit Recycling in Quantum Computation
Experimental realization of Shor’s quantum factoring algorithm using qubit recycling
Ran Chu
Introduce quantum computing and quantum recycling with “Experimental realization of Shor’s quantum factoring algorithm using qubit recycling”

2. What’s quantum computing?
The idea of quantum computing is using quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.
Universal gate: 2 qubits(CU) local
unitary transformations
CLASSICAL CIRCUITS => gate. quantum operation => unitary transformations
CCNOT (4 qubits, universal gate in classical computer) => any 4*4 unitary matrix (2 qubits) : can be proved with Deutch gate, 8*8 matrix (4 qubits), which can be constructed with any 2 qubits unitary transformations

3. Quantum Gates
CLASSICAL CIRCUITS => gate. quantum operation => unitary transformations
CCNOT (4 qubits, universal gate in classical computer) => any 4*4 unitary matrix (2 qubits) : can be proved with Deutch gate, 8*8 matrix (4 qubits), which can be constructed with any 2 qubits unitary transformations

4. Quantum Fourier Transform
Classical Fast Fourier Transform:
DFT:
Quantum Fourier Transform: gates
Shor’s algorithm is designed to work for even orders only. gcd(x r 2 ± 1, N).
A single pure qubit, together with a collection
of log2N qubits in an arbitrary mixed state, is sufficient to implement Shor’s factorization algorithm efficiently
64 mod 21 = 1

5. Algebra

6. Play with Quantum Fourier Transform
The number of steps any classical computer requires in order to
find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present1. Factoring
large integers is therefore conjectured to be intractable classically,
an observation underlying the security of widely used cryptographic codes1,2. Quantum computers3, however, could factor
integers in only polynomial time, using Shor's quantum factoring
algorithm4
An algorithm is said to be solvable in polynomial time if the number of steps required to complete the algorithm for a given input is O(n^k) for some nonnegative integer k, where n is the complexity of the input. Polynomial-time algorithms are said to be "fast."

7. Shor’s Quantum Factoring Algorithm
Goal： determine the prime factors of an odd integer N
Example: N =21 (21 = 3 * 7 = 1* 21)
1, choses a co-prime of N=21: x =4
2, find the order relates 4 to 21 : 43 mod 21 = 64 mod 21 = 1
3, gcd(43/2±1, 21) = gcd(8+1,21)=3 or gcd(8-1,21)=7
Requirements:
1,Shor’s algorithm is designed to work for even orders only. gcd(x^r/2 ± 1, N). 2,A single pure qubit, together with a collection of log_{2}(N) qubits in an arbitrary mixed state, is sufficient to implement Shor’s factorization algorithm efficiently.[S.Parker 2000]
By repeating O(log log r) times
Shor’s algorithm is designed to work for even orders only. gcd(x r 2 ± 1, N).
A single pure qubit, together with a collection
of log2N qubits in an arbitrary mixed state, is sufficient to implement Shor’s factorization algorithm efficiently
64 mod 21 = 1

8. Compiling Shor’s Quantum Factoring Algorithm
Vandersypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S., Sherwood, M. H., & Chuang, I. L. (2001). Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature, 414(6866), 883–887.
H in a) is a Hadamard gate. It works on a single qubit.

9. Compiling the Algorithm
H in a) is a Hadamard gate. It works on a single qubit.

10. Qubit Recycling

11. Method
Has only 1 control qubit. => uniform distribution

12. Outcome The algorithmic output is distinguishable from noise.
The total number of qubits is one-third of that required in the standard protocol.

13. Limitations
The quantum circuits only valid for factoring N = 21 with x =4.
It's reasonable to double whether it can be consider as implementation of Shor's algorithm. [Smolin 2013]
Qubit recycling is valid only when the control qubits can be operated on work register bit by bit.

14. Reference
Martín-López, E., Laing, A., Lawson, T., Alvarez, R., Zhou, X.-Q., & O’Brien, J. L. (2012). Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nature Photonics, 6(11), 773–776.
Parker, S., & Plenio, M. B. (2000). Efficient Factorization with a Single Pure Qubit and logN Mixed Qubits S. Physical Review Letters, 85(14), 3049–
Vandersypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S., Sherwood, M. H., & Chuang, I. L. (2001). Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature, 414(6866), 883–887.
Politi, A., Matthews, J. C. F., & O’Brien, J. L. (2009). Shor’s Quantum Factoring Algorithm on a Photonic Chip. Science, 325(5945), 1221–
Smolin, J. A., Smith, G., & Vargo, A. (2013). Oversimplifying quantum factoring. Nature, 499(7457), 163–165.

15. Politi, A. , Matthews, J. C. F. , & O’Brien, J. L. (2009)
Politi, A., Matthews, J. C. F., & O’Brien, J. L. (2009). Shor’s Quantum Factoring Algorithm on a Photonic Chip. Science, 325(5945), 1221–

16. Photon source
type I spontaneous parametric downconversion source