Simulating classical circuits with quantum circuits. The complexity class Reversible-P. Universal gate set for reversible circuits P = Reversible-P

Transcription

1 Quantum Circuits Based on notes by John Watrous and also Sco8 Aaronson: John Preskill:

2 Outline Simulating classical circuits with quantum circuits The complexity class Reversible-P Universal gate set for reversible circuits P = Reversible-P The complexity class QBP Universal gate set for quantum circuits BPP QBP PSPACE

3 Review: Toffoli gates If a =1 and b= 1, NOT c Otherwise c

4 Review: classical à reversible circuits AND gates NOT gates (can also be simulated by Toffoli gates) Fanout

5 Review: classical à reversible circuits All of the operations needed for simulating classical circuits map classical states to classical states Such operations can be represented by permutation matrices Circuits or operations that always map classical states to classical states are called reversible operations or circuits

6 Exercise Output What is the superposiion at the blue do8ed line? What is the probability of measuring 1?

7 Exercise Output What is the superposiion at the blue do8ed line? What is the probability of measuring 1?

8 Classical à reversible circuits

9 Classical à reversible circuits

10 Classical à reversible circuits We can clean up garbage bits by restoring the ancilla bits using two tricks: Copy the output using controlled NOT gates, then Use the mirror image of R to restore the original input

11 Classical à reversible circuits

12 Classical à reversible circuits

13 Classical à reversible circuits

14 Classical à reversible circuits (ignoring ancilla bits)

15 SimulaIng classical circuits: summary If f: {0,1} n à {0,1} m can be computed by classical circuit C, then our simulaion procedure generates a reversible circuit S C that saisfies The size of S C is polynomial in the size of C

16 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity Acceptance

17 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates

18 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity

19 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity {C n n 0} is a uniform family of poly-size Boolean circuits if C n has n input bits and one designated output bit there is a polynomial Ime algorithm that can produce C n, given 1 n as input

20 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity: poly(n)-ime algorithm to produce C n

21 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity: poly(n)-ime algorithm to produce C n Acceptance: designated output bit is 1

22 Recall: circuit complexity classes Input: binary string x Universal gate sets (bases): NOT, AND gates Uniformity: poly(n)-ime algorithm to produce C n Acceptance: designated output bit is 1 A language is in P iff there is a uniform family {C n } of poly-size circuits such that If x L then C n accepts input x If x L then C n does not accept input x

23 BPP as a circuit complexity class Input: binary string x and coin flip bits Universal gate sets (bases): NOT, AND gates Uniformity: poly(n)-ime algorithm to produce C n Acceptance: designated output bit is 1

24 BPP as a circuit complexity class Input: binary string x and coin flip bits Universal gate sets (bases): NOT, AND gates Uniformity: poly(n)-ime algorithm to produce C n Acceptance: designated output bit is 1 A language is in BPP iff there is a uniform family {C n } of poly-size circuits such that If x L then C n accepts input x with probability 2/3 If x L then C n accepts input x with probability 1/3

25 Reversible circuit complexity classes

26 Reversible circuit complexity classes Ini?al state Universal gate sets (bases) Uniformity Acceptance

27 Reversible circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases) Uniformity Acceptance

28 Reversible circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli

29 Reversible circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli Uniformity: as for classical circuits Acceptance: an output bit is measured once all operaions are applied; a measurement of 1 denotes accept and 0 denotes reject

30 Reversible circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli Uniformity: as for classical circuits Acceptance: output bit is 1

31 Reversible circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli Uniformity: as for classical circuits Acceptance: output bit is 1 A language is in Reversible-P iff there is a uniform family {C n } of poly-size reversible circuits such that If x L then C n accepts input x If x L then C n does not accept input x

32 Reversible-P = P Reversible-P P: follows since any gates (including Toffoli gates) can be simulated by NOT and AND gates P Reversible-P: follows from our simulaion of classical circuits by reversible circuits

33 Quantum circuit complexity classes

34 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits)

35 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases)

36 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases) For quantum computaion, we can work with realvalued sets of quantum gates/unitary operators Such a basis set is universal for quantum computaion if any real unitary operator can be approximated with arbitrary precision by a circuit involving only the basis gates

37 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases)

38 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases) Yaoyun Shi (2002) showed that Toffoli and Hadamard gates form a universal set Solovay-Kitaev: Any universal set of gates can simulate any other universal set with at most a polynomial increase in the number of gates

39 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli, Hadamard Uniformity: as for classical circuits

40 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli, Hadamard Uniformity: as for classical circuits Acceptance

41 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli, Hadamard Uniformity: as for classical circuits Acceptance: An output bit is measured once all operaions have been applied; a measurement of 1 denotes accept and 0 denotes reject. In quantum circuits, the measurement is probabilis?c

42 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli, Hadamard Uniformity: as for classical circuits Acceptance: output bit is 1

43 Quantum circuit complexity classes Ini?al state: x (actual input plus ancilla bits) Universal gate sets (bases): Toffoli, Hadamard Uniformity: as for classical circuits Acceptance: output bit is 1 A language is in BQP iff there is a uniform family {C n } of poly-size quantum circuits such that If x L then C n accepts input x with probability 2/3 If x L then C n accepts input x with probability 1/3

44 BQP versus tradiional classes A first step towards understanding the power of quantum compuing is to situate BQP in our zoo of complexity classes

45 BPP is contained in BQP Let C be a BPP circuit: Input: binary string x plus coin flips Gate set: NOT and AND Accept: if output bit is 1

46 BPP is contained in BQP Obtain an equivalent quantum circuit as follows: Apply the classical à reversible gate conversion, (adding the needed ancilla bits) Replace each coin flip input by a qubit that is iniialized to 0 Add a Hadamard gate as the first operaion that is applied to each coin flip qubit Why does this work?

47 Evidence that BPP BQP Factoring: given a posiive integer N, output a prime factorizaion of N The best classical algorithms take Ime exponenial in log N Shor s algorithm takes Ime O((log N) 3 ) Prior to Shor s algorithm, Simon s algorithm was an interesing example of the power of quantum computaion

48 Simon s Problem Let f: {0,1} n à {0,1} n be such that for some s in {0,1} n for all x,y in {0,1} n f(x)=f(y) if and only if x y {0 n, s} Suppose that we have a "black box for f, and can query f(x) for any x. How many queries are needed to find s with high probability? Ω ( (2 n ) queries needed classically O(n) queries are needed with quantum operaions

49 BQP is in PSPACE Not hard to see that BQP is in EXP To show BQP is in PSPACE, adapt argument that BPP is in PSPACE

50 Summary BPP QBP PSPACE It s conjectured that BPP BQP: the fact that Factoring is in BQP is strong evidence

51 Next class Upper bounds on BQP BQP is in PSPACE BQP is in PP Overview of Simon s algorithm