QWIRE Practice: Formal Verification of Quantum Circuits in Coq

Transcription

1 1 / 29 QWIRE Practice: Formal Verification of Quantum Circuits in Coq Robert Rand, Jennifer Paykin, Steve Zdancewic University of Pennsylvania Quantum Physics and Logic, 2017

2 2 / 29 QWIRE A high-level language for generating quantum circuits following Quipper and LIQUi A minimal set of primitives for circuit construction Programs are guaranteed to correspond to realizable quantum computations using linear types, as in the Quantum Lambda Calculus

3 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

4 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

5 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

6 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

7 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

8 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

9 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

10 3 / 29 Preparing a Bell state 0 H x 0 y Definition bell00 : Box One (Qubit Qubit). box () gate x gate y gate x gate (x,y) output (x,y) Defined.

11 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

12 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

13 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

14 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

15 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

16 4 / 29 Quantum Teleportation alice bob bell00 0 H H meas meas 0 X Z Definition teleport : Box Qubit Qubit. box t let (p,q) unbox bell00 (); let (a,b) unbox alice (t,p); let t' unbox bob (a,b,q); output t' Defined.

17 5 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort.

18 5 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort.

19 6 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort. Definition delete : Box Qubit One. box q gate q ; output (). Abort.

20 6 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort. Definition delete : Box Qubit One. box q gate q ; output (). Abort.

21 7 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort. Definition delete : Box Qubit One. box q gate q ; output (). Abort. Definition recall : Box Qubit Qubit. box q gate p ; output q. Abort.

22 7 / 29 Non-Quantum Circuits Definition clone : Box Qubit (Qubit Qubit). box q output (q,q). Abort. Definition delete : Box Qubit One. box q gate q ; output (). Abort. Definition recall : Box Qubit Qubit. box q gate p ; output q. Abort.

23 8 / 29 Linear Types are not Enough Sources of Errors Provide the wrong argument to a phase shift gate. Apply a unitary to the wrong wire, or vice-versa. Forget to uncompute or measure-discard certain qubits Classically compute the wrong circuit.

24 9 / 29 Approaches Simulation Debugging Unit Testing Math + Concentration

25 9 / 29 Approaches Simulation X Debugging Unit Testing Math + Concentration

26 9 / 29 Approaches Simulation X Debugging XX Unit Testing Math + Concentration

27 9 / 29 Approaches Simulation X Debugging XX Unit Testing $$$ Math + Concentration

28 9 / 29 Approaches Simulation X Debugging XX Unit Testing $$$ Math + Concentration?

29 10 / 29 The Coq Proof Assistant A programming language An interactive theorem prover

30 11 / 29 The Coq Proof Assistant Accomplishments Mathematics Feit-Thompson Theorem Four Color Theorem Programming CompCert Compiler CertiKOS

31 12 / 29 Coq + QWIRE A circuit description language A density matrix library A denote function from circuits to superoperators Integrated proofs of program correctness

32 13 / 29 Denoting Programs Definition denote_unitary : Unitary Matrix. Definition denote_gate : Gate Superoperator. Definition denote_circuit : Circuit Superoperator. Class Denote source target := {denote : source target}. Notation " s " := (denote s) (at level 10).

33 14 / 29 Denoting a Coin Flip 0 H meas Definition coin_flip : Box One Bit. box () gate q gate q gate b output b. Defined.

34 15 / 29 Denoting a Coin Flip 0 H meas Lemma fair_toss : coin_flip [1] = [ [1/2, 0] [0, 1/2] ].

35 16 / 29 Denoting a Coin Flip 0 H meas = 0 0 H 0 0 H H 0 0 H 1 1

36 16 / 29 Denoting a Coin Flip 0 H meas = ( ) ( ) ( ) ( ) ( ) 2 + ( ) ( ) ( ) ( ) ( ) 2

37 16 / 29 Denoting a Coin Flip 0 H meas = ( ) + ( ) 2

38 16 / 29 Denoting a Coin Flip 0 H meas = ( ) 2

39 Verified 17 / 29

40 18 / 29 Denoting Coin Flips 0 H meas 0 H meas 0 H meas (* Generates a binary number between 0 and 2 n *) Definition uniform (n : N) : Box (n One) (n Bit) := parallel_copy n coin_flip

41 19 / 29 Denoting Coin Flips 0 H meas 0 H meas 0 H meas ( ) ( ) ( ) 2

42 Denoting Coin Flips 0 H meas 0 H meas 0 H meas 2 n n n 19 / 29

43 A Biased Coin Base (m = 0) Recursive (m = n + 1) biased_coin n 1 0 H meas Fixpoint biased_coin (m : N) : Box One Bit. box () match m with 0 gate x output x n+1 let c unbox (biased coin n) (); gate q gate (c,q) bit_ctrl gate () gate b output b end. Defined. 20 / 29

44 A Biased Coin Base (m = 0) Recursive (m = n + 1) biased_coin n 1 0 H meas Fixpoint biased_coin (m : N) : Box One Bit. box () match m with 0 gate x output x n+1 let c unbox (biased coin n) (); gate q gate (c,q) bit_ctrl gate () gate b output b end. Defined. 20 / 29

45 A Biased Coin Base (m = 0) Recursive (m = n + 1) biased_coin n 1 0 H meas Fixpoint biased_coin (m : N) : Box One Bit. box () match m with 0 gate x output x n+1 let c unbox (biased coin n) (); gate q gate (c,q) bit_ctrl gate () gate b output b end. Defined. 20 / 29

46 A Biased Coin Base (m = 0) Recursive (m = n + 1) biased_coin n 1 0 H meas Fixpoint biased_coin (m : N) : Box One Bit. box () match m with 0 gate x output x n+1 let c unbox (biased coin n) (); gate q gate (c,q) bit_ctrl gate () gate b output b end. Defined. 20 / 29

47 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas

48 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas n n+1

49 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas n n+1 biased coin 0 = ( )

50 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas n n+1 biased coin 0 = ( ) biased coin (n+1) = meas 2 (ctrl-h( biased coin n 0 0 ))

51 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas n n+1 biased coin 0 = ( ) biased coin (n+1) = meas 2 (ctrl-h( biased coin n 0 0 )) = meas 2 (ctrl-h(( n 0 0 ) 0 0 )) 1 2 n

52 21 / 29 A Biased Coin: Induction biased_coin n 0 H meas n n+1 biased coin 0 = ( ) biased coin (n+1) = meas 2 (ctrl-h( biased coin n 0 0 )) = meas 2 (ctrl-h(( n 0 = ( n ) 1 2 n+1 0 ) 0 0 )) 1 2 n

53 22 / 29 Denoting Unitaries U U Definition unitary_transpose (n : N) (U : Unitary n) : Box (n Qubit) (n Qubit). box ρ gate ρ gate ρ transpose output ρ. Defined.

54 23 / 29 Denoting Unitaries U U Lemma unitary_trans_id : (n : N) (U : Unitary n) (ρ : Density_Matrix n), unitary_transpose U ρ = ρ.

55 24 / 29 Denoting Unitaries U U unitary transpose ρ = U (UρU )(U )

56 24 / 29 Denoting Unitaries U U unitary transpose ρ = U (UρU )U

57 24 / 29 Denoting Unitaries U U unitary transpose ρ = (U U)ρ(U U)

58 24 / 29 Denoting Unitaries U U unitary transpose ρ = IρI

59 24 / 29 Denoting Unitaries U U unitary transpose ρ = ρ

60 25 / 29 Quantum Teleportation H meas 0 H meas 0 X Z Lemma teleport_id : (ρ : Density_Matrix 1), teleport ρ = ρ.

61 26 / 29 Entangled Quantum Teleportation H meas 0 H meas 0 X Z Lemma teleport_id' : (n : N) (ρ : Density_Matrix (n+1)), in_parallel (id n) teleport ρ = ρ.

62 27 / 29 Why QWIRE? Linearly typed circuits guarantee quantum realizability Dependent types allow precise specification of circuit families Embedded in Coq for classical control Denotational semantics for verified programming Categorical semantics (Renella and Staton, 2017)

63 28 / 29 Future Work Verified complex programs (QFT, Shor, Grover) Reversible circuits and verified compilation as in ReVer (Amy et al., 2017) Quantum oracles proven to correspond to classical functions

64 29 / 29 FIN T H A N K Y O U