Communicating Quantum Processes

1. Communicating Quantum Processes Simon Gay Department of Computing Science University of Glasgow Rajagopal Nagarajan Department of Computer Science University of Warwick This work was presented at the Second International Workshop on Quantum Programming Languages, Turku, Finland, July 2004, and at the ACM Symposium on Principles of Programming Languages (POPL), Long Beach, California, January 2005.

2. Overview Quantum computing has the potential to solve some hard problems efficiently. Only a few algorithms are known and at present there are no techniques for generic improvement of algorithmic performance. Physical implementations of quantum computers of useful size are a long way in the future. Shor’s algorithm (1994) can efficiently factorize large integers. A practical implementation of Shor’s algorithm would kill much of our present cryptographic technology. Surrey Seminar: Communicating Quantum Processes

3. Overview Quantum cryptographic techniques are secure even in the presence of quantum computers. Quantum cryptography is much easier to implement than quantum computing. City-scale demonstrations have taken place, and components are commercially available. Quantum cryptography (and other quantum communication systems) will definitely be an important practical technology in the near future. Surrey Seminar: Communicating Quantum Processes

4. Overview Although quantum cryptographic protocols have been proved mathematically to be secure, we believe that there is a need for verification of systems which combine quantum communication with classical computation and communication. Computer science has a range of theories, techniques and tools which have been successfully used to verify classical communication and cryptographic systems. Formal modelling languages, specification logics, type theory, model-checking tools, automated theorem-proving, ... Our research programme: to apply these techniques to combined quantum/classical systems. Surrey Seminar: Communicating Quantum Processes

5. Outline Background on quantum computing. Quantum communication: dense coding. The language CQP (pi calculus + quantum operations). illustration through the above examples formal semantics and type system Future work. Surrey Seminar: Communicating Quantum Processes

6. Qubits In classical computing the fundamental unit of information is the bit. The value of a bit is either 0 or 1. In quantum computing the fundamental unit of information is the quantum bit or qubit. A qubit has two basis states: |0 and |1 (physics notation). A general state of a qubit is |0 + |1 where  and  are complex numbers, usually normalized so that If both  and  are non-zero then the state is said to be a superposition. Surrey Seminar: Communicating Quantum Processes

7. Measurement A qubit can be measured to produce a classical value. If a qubit is in state |0 + |1 then the outcome of a measurement is probabilistic. With probability the result is 0 and the qubit enters state |0. With probability the result is 1 and the qubit enters state |1. Example: measure a qubit which is in state with probability 0.5 the result is 0 and the qubit enters state |0 with probability 0.5 the result is 1 and the qubit enters state |1 Surrey Seminar: Communicating Quantum Processes

8. Systems of Multiple Qubits For example, consider a system of 2 qubits. There are 4 basis states: |0|0|0|1 |1|0 |1|1 A general state is a superposition of the basis states: where A measurement of both qubits has the following outcome: With probability the result is 0 and the new state is |0|0. With probability the result is 1 and the new state is |0|1. With probability the result is 2 and the new state is |1|0. With probability the result is 3 and the new state is |1|1. Surrey Seminar: Communicating Quantum Processes

9. Systems of Multiple Qubits For example, consider a system of 2 qubits. There are 4 basis states: |0|0|0|1 |1|0 |1|1 A general state is a superposition of the basis states: where A measurement of the first qubit has the following outcome: With prob. result 0 and new state With prob. result 1 and new state Surrey Seminar: Communicating Quantum Processes

10. Quantum Parallelism Suppose we have a function f : {0,1}  {0,1} from which we can derive a transformation F on qubits, such that Transformations extend linearly to superpositions, so For a function on n bits we can compute function applications simultaneously. Magic! This is quantum parallelism and is the basis for the simplistic claim that quantum computing can deliver exponential efficiency gains for general problems. BUT the only way to extract information is to measure, so we get either f(0) or f(1) and we don’t even know which it is! Surrey Seminar: Communicating Quantum Processes

11. Deutsch’s Problem Suppose we have a black box which computes an unknown function f : {0,1}  {0,1} , and we want to know whether or not f is a constant function. Classically we can’t do better than calculating f(0) and f(1) and comparing the results. But suppose we can ask for a quantum version of the black box. It’s not reasonable to expect F such that in general, because quantum theory requires F to be invertible and f might not be invertible. Surrey Seminar: Communicating Quantum Processes

12. Deutsch’s Problem It is reasonable to ask for a transformation F such that so let’s assume that this is the quantum version of the black box. We will make use of the superposed qubit states and apply F to the state Surrey Seminar: Communicating Quantum Processes

13. Deutsch’s Problem If f(x)=0 then If f(x)=1 then Therefore Surrey Seminar: Communicating Quantum Processes

14. Deutsch’s Problem Therefore if f(0) = f(1) if f(0)  f(1) There is a transformation H such that so applying H to the first qubit gives if f(0) = f(1) if f(0)  f(1) Surrey Seminar: Communicating Quantum Processes

15. Deutsch’s Problem We end up with the following state: if f(0) = f(1) if f(0)  f(1) and now measuring the first qubit gives us the desired information about f , and we only used the quantum black box once. Quantum parallelism was used to calculate f(0) and f(1) ; a global property of f ended up being encoded in a single place so that it could be extracted by a measurement. Developing quantum algorithms for interesting problems seems to be very difficult. Surrey Seminar: Communicating Quantum Processes

16. Entanglement A state of n qubits which is not of the form is said to be entangled. The simplest example: 2 qubits in state Suppose we measure the first qubit from this state: With probability 0.5 the result is 0 and the new state is |0|0. With probability 0.5 the result is 1 and the new state is |1|1. In both cases, the two qubits now have the same state. Measuring the second qubit is guaranteed to give the same result as the first measurement, even if the qubits are physically separated by any distance. Surrey Seminar: Communicating Quantum Processes

17. Transformation / Evolution Evolution of a quantum system is described by a Hermitian operator. Concretely, for a system of n qubits, an operator is a matrix A such that where A* is the conjugate transpose. Example: The Hadamard operator H is defined by Represent by then Surrey Seminar: Communicating Quantum Processes

18. Controlled Not The controlled not operator applies to 2 qubits. Using H and CNot we can prepare an entangled state: Surrey Seminar: Communicating Quantum Processes

19. Pauli Operators The Pauli operators will be useful in a moment. They apply to single qubits. Surrey Seminar: Communicating Quantum Processes

20. The No Cloning Theorem There is no unitary transformation U such that for all Ψ ( s is some standard initial state for the second qubit). Proof: (Dieks; Wootters & Zurek; 1982) Take inner product: Therefore either or So φ and ψ are either orthogonal or identical. Surrey Seminar: Communicating Quantum Processes

21. Dense Coding A protocol allowing 2 classical bits of information to be transmitted by sending 1 qubit and making use of entanglement. Alice and Bob share an entangled pair of qubits: x, y. Alice wishes to send n (0  n  3) to Bob. There is a quantum channel q linking them. Alice Bob x y (0  n  3) can transmit qubits q Surrey Seminar: Communicating Quantum Processes

22. Dense Coding A protocol allowing 2 classical bits of information to be transmitted by sending 1 qubit and making use of entanglement. Alice and Bob share an entangled pair of qubits: x, y. Alice wishes to send n (0  n  3) to Bob. There is a quantum channel q linking them. Alice(x:Qbit, q:^[Qbit], n:0..3) = ... Bob(y:Qbit, q:^[Qbit]) = ... System(x:Qbit, y:Qbit, n:0..3) = (new q:^[Qbit])( Alice(x,q,n) | Bob(y,q) ) Surrey Seminar: Communicating Quantum Processes

23. Dense Coding Alice selects a Pauli transformation corresponding to n, applies it to x, and sends x to Bob. Alice Bob x y send x to Bob x Alice(x:Qbit, q:^[Qbit], n:0..3) = {x *= n} . q![x] . Stop Surrey Seminar: Communicating Quantum Processes

24. Dense Coding Alice selects a Pauli transformation corresponding to n, applies it to x, and sends x to Bob. Alice Bob x y eg. n=1 send x to Bob x Alice(x:Qbit, q:^[Qbit], n:0..3) = {x *= n} . q![x] . Stop Surrey Seminar: Communicating Quantum Processes

25. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Alice Bob x y send x to Bob x x, y CNot x, y H measure x,y n x, y Surrey Seminar: Communicating Quantum Processes

26. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Alice Bob x y eg. n=1 send x to Bob x CNot x, y H measure x,y n x, y Surrey Seminar: Communicating Quantum Processes

27. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Alice Bob x y eg. n=1 send x to Bob x CNot H measure x,y n x, y Surrey Seminar: Communicating Quantum Processes

28. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Alice Bob x y eg. n=1 send x to Bob x CNot H measure x,y n Surrey Seminar: Communicating Quantum Processes

29. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Alice Bob x y eg. n=1 send x to Bob x CNot H measure x,y 1 Surrey Seminar: Communicating Quantum Processes

30. Dense Coding Bob receives x, applies CNot to x,y, applies H to x, then measures x,y. The result uniquely determines n. Bob(y:Qbit, q:^[Qbit]) = q?[z:Qbit] . {z,y *= CNot} . {z *= H} . Use(measure z,y) Surrey Seminar: Communicating Quantum Processes

31. Dense Coding Alice(x:Qbit, q:^[Qbit], n:0..3) = {x *= n} . q![x] . Stop Bob(y:Qbit, q:^[Qbit]) = q?[z:Qbit] . {z,y *= CNot} . {z *= H} . Use(measure z,y) System(x:Qbit, y:Qbit, n:0..3) = (new q:^[Qbit])( Alice(x,q,n) | Bob(y,q) ) System is parameterized by x and y (and we must assume that they are entangled). Surrey Seminar: Communicating Quantum Processes

32. Dense Coding with Entanglement Server We can make the system more self-contained: Alice and Bob receive their entangled qubits from a third process. (EPR source or entanglement server) Alice’(s:^[Qbit], q:^[Qbit], n:0..3) = s?[x:Qbit] . Alice(x,q,n) Bob’(t:^[Qbit], q:^[Qbit]) = t?[y:Qbit] . Bob(y,q) Source(s:^[Qbit], t:^[Qbit]) = (qbit x,y)({x *= H} . {x,y *= CNot} . s![x] . t![y] . Stop) System(n:0..3) = (new q, s, t:^[Qbit])( Alice’(s,q,n) | Bob’(t,q) | Source(s,t) ) Surrey Seminar: Communicating Quantum Processes

33. The Language CQP This example has introduced our language CQP. Defining a formal language for describing protocols is a valuable step. We have also defined a formal operational semantics. It is based on the pi calculus, with the addition of global state (for the qubits) and probabilistic reductions (arising from measurements). CQP has a static type system which, in addition to checking correct use of values and channels, guarantees that qubits are not syntactically copied (because of the No Cloning Theorem). Surrey Seminar: Communicating Quantum Processes

34. Reduction in CQP Communication Surrey Seminar: Communicating Quantum Processes

35. Reduction in CQP Measurement Surrey Seminar: Communicating Quantum Processes

36. Typechecking for No Cloning Suppose that Alice continues with some behaviour, after sending the qubit to Bob. Alice(x:Qbit, q:^[Qbit], n:0..3) = {x *= n} . q![x] . Continue The type system of CQP guarantees that Continue does not refer to x. Technically: the fact that x is sent along q means that Continue is typechecked in an environment which does not contain x. The details are based on several pi calculus type systems, going back to Kobayashi, Pierce & Turner (1996). Surrey Seminar: Communicating Quantum Processes

37. Teleportation Alice and Bob share an entangled pair x,y of qubits. Alice has a qubit z in an unknown state, and she wishes to transmit this state to Bob (perhaps the physical form of z is not suitable for a direct transfer). Alice applies CNot to z,x and then applies H to z. Then she measures z,x to yield a two-bit classical value which she sends to Bob on a channel c. Bob uses this value to select a Pauli transformation and applies it to y. The result is that the final state of y is the same as the initial state of z. Surrey Seminar: Communicating Quantum Processes

38. Teleportation in CQP Alice(x:Qbit, c:^[0..3], z:Qbit) = {z,x *= CNot} . {z *= H} . c![measure z,x] . Stop Bob(y:Qbit, c:^[0..3]) = c?[r:0..3] . {y *= r} . Use(y) System(x:Qbit, y:Qbit, z:Qbit) = (new c:^[0..3])( Alice(x,c,z) | Bob(y,c) ) Surrey Seminar: Communicating Quantum Processes

39. How Teleportation Works x and y are entangled: z has some general state: Overall state: Surrey Seminar: Communicating Quantum Processes

40. How Teleportation Works z,x *= CNot z *= H Surrey Seminar: Communicating Quantum Processes

41. How Teleportation Works measure z,x y *= 0 result = 0 y *= 1 result = 1 y *= 2 result = 2 y *= 3 result = 3 In each case the final state of y is the same as the initial state of z, (and there is no entanglement between z,x and y). Surrey Seminar: Communicating Quantum Processes

42. Related Work There are several papers on quantum programming languages and our syntax has been influenced in particular by Selinger’s QPL (2003). Jorrand & Lalire (2004) have defined a quantum process algebra (QPA), which is quite similar to CQP in several ways (there has now been considerable influence in both directions). The type system is one of our distinctive features. Surrey Seminar: Communicating Quantum Processes

43. Future Work The purpose of defining CQP is to support formal reasoning about quantum systems. There are several directions to explore. Verification tools: we arenow working with PRISM models constructed in its (low-level) language. In the future we aim to construct PRISM models by translation from CQP. Translation into non-probabilistic model-checking systems may also be useful, for exploration of possibilities before analysis of probabilities. Translation into some existing simulation framework may also be worthwhile. Surrey Seminar: Communicating Quantum Processes

44. Future Work Logics for specification: for both automatic and non-automatic verification. Example: a group at Technical University of Lisbon is developing a Hoare-style logic for quantum systems, but they have no formal syntax for describing systems. { pre } P { post } ad hoc description of a system Type-theoretic methods: can we get more from a type system? For example: can we develop type systems for high-level properties such as secrecy? (This is known for classical security protocols.) Surrey Seminar: Communicating Quantum Processes