Quantum computing and mathematical research. Chi-Kwong Li The College of William and Mary

Transcription

1 and mathematical research The College of William and Mary

2 Classical computing

3 Classical computing Hardware - Beads and bars.

4 Classical computing Hardware - Beads and bars. Input - Using finger skill to change the states of the device.

5 Classical computing Hardware - Beads and bars. Input - Using finger skill to change the states of the device. Processor - Mechanical process with clever algorithms based on elementary arithmetic rules.

6 Classical computing Hardware - Beads and bars. Input - Using finger skill to change the states of the device. Processor - Mechanical process with clever algorithms based on elementary arithmetic rules. Output - Beads and bars, then recorded by brush and ink.

7 Modern computing

8 Modern computing Hardware - Mechanical/electronic/integrated circuits.

9 Modern computing Hardware - Mechanical/electronic/integrated circuits. Input - Punch cards, keyboards, scanners, sounds, etc. all converted to binary bits - (0, 1) sequences.

10 Modern computing Hardware - Mechanical/electronic/integrated circuits. Input - Punch cards, keyboards, scanners, sounds, etc. all converted to binary bits - (0, 1) sequences. Processor - Manipulations of (0, 1) sequences using Boolean logic.

11 Modern computing 0 0 = = = = 1 Hardware - Mechanical/electronic/integrated circuits. Input - Punch cards, keyboards, scanners, sounds, etc. all converted to binary bits - (0, 1) sequences. Processor - Manipulations of (0, 1) sequences using Boolean logic.

12 Modern computing 0 0 = = = = 1 Hardware - Mechanical/electronic/integrated circuits. Input - Punch cards, keyboards, scanners, sounds, etc. all converted to binary bits - (0, 1) sequences. Processor - Manipulations of (0, 1) sequences using Boolean logic. Output - (0, 1) sequences realized as visual images, which can be viewed or printed.

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14 Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc.

15 Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Input - Quantum states in a specific form - Qubit (Quantum bit).

16 Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Input - Quantum states in a specific form - Qubit (Quantum bit). Processor - Provide suitable environment for the quantum system of qubits to evolve.

17 Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Input - Quantum states in a specific form - Qubit (Quantum bit). Processor - Provide suitable environment for the quantum system of qubits to evolve. Output - Measurement of the resulting quantum states.

18 Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Input - Quantum states in a specific form - Qubit (Quantum bit). Processor - Provide suitable environment for the quantum system of qubits to evolve. Output - Measurement of the resulting quantum states. All these require the understanding of mathematics, physics, chemistry, computer sciences, engineering, etc.

19 Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment.

20 Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment. Dr. Quantum - Entanglement.

21 Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment. Dr. Quantum - Entanglement. Quantum teleportation.

22 Mathematical challenges How to control the (initial) quantum states?

23 Mathematical challenges How to control the (initial) quantum states? How to create the appropriate environment for the quantum mechanical system to evolve without observing?

24 Mathematical challenges How to control the (initial) quantum states? How to create the appropriate environment for the quantum mechanical system to evolve without observing? How to fight decoherence (the interaction of the system and the external environment)?

25 Mathematical challenges How to control the (initial) quantum states? How to create the appropriate environment for the quantum mechanical system to evolve without observing? How to fight decoherence (the interaction of the system and the external environment)? How to use the phenomena of superposition and entanglement effectively to design quantum algorithms.

26 Quantum states and quantum bits Suppose a quantum system have two (discrete) measurable physical sates, say, up spin and down spin of a particle.

27 Quantum states and quantum bits Suppose a quantum system have two (discrete) measurable physical sates, say, up spin and down spin of a particle. They will be represented by 0 = ( ) 1 0 and 1 = ( ) 0. 1

28 Quantum states and quantum bits Suppose a quantum system have two (discrete) measurable physical sates, say, up spin and down spin of a particle. They will be represented by 0 = ( ) 1 0 and 1 = ( ) 0. 1 Inside the black box, the vector state may be in superposition state (the famous Schrödinger cat which could be half alive and half dead) represented by ( ) α v = ψ = α 0 + β 1 = C 2, α 2 + β 2 = 1. β

29 Quantum states and quantum bits Suppose a quantum system have two (discrete) measurable physical sates, say, up spin and down spin of a particle. They will be represented by 0 = ( ) 1 0 and 1 = ( ) 0. 1 Inside the black box, the vector state may be in superposition state (the famous Schrödinger cat which could be half alive and half dead) represented by ( ) α v = ψ = α 0 + β 1 = C 2, α 2 + β 2 = 1. β In quantum mechanics/computing, one uses such a basic quantum state/ quantum bit (qubit).

30 In fact, it is more convenient to represent the quantum state ψ as a rank one orthogonal projection: ( ) 1 + z x + iy Q = vv = ψ ψ = 1 2 x iy 1 z with x, y, z R such that x 2 + y 2 + z 2 = 1.

31 In fact, it is more convenient to represent the quantum state ψ as a rank one orthogonal projection: ( ) 1 + z x + iy Q = vv = ψ ψ = 1 2 x iy 1 z with x, y, z R such that x 2 + y 2 + z 2 = 1. There is a Bloch sphere representation of a qubit

32 Quantum gates and quantum operations The state of k qubits is represented as the tensor product of k 2 2 matrices Q 1 Q k.

33 Quantum gates and quantum operations The state of k qubits is represented as the tensor product of k 2 2 matrices Q 1 Q k. To accommodate various quantum effects, one considers 2 k 2 k density matrices, i.e., trace one positive semidefinite Hermitian matrices.

34 All quantum gates and quantum evolutions (for a closed system) are unitary similarity transforms of the density matrices representing the states, i.e., A(t) U(t)A(0)U(t) for unitaries U(t).

35 All quantum gates and quantum evolutions (for a closed system) are unitary similarity transforms of the density matrices representing the states, i.e., A(t) U(t)A(0)U(t) for unitaries U(t). All quantum operations, quantum channels, quantum measurement, etc. are (trace preserving) completely positive linear map Φ

36 All quantum gates and quantum evolutions (for a closed system) are unitary similarity transforms of the density matrices representing the states, i.e., A(t) U(t)A(0)U(t) for unitaries U(t). All quantum operations, quantum channels, quantum measurement, etc. are (trace preserving) completely positive linear map Φ admitting the operator sum representation A r X i AXj, j=1 with r j=1 X j X j = I in case Φ is trace preserving.

37 Some sample matrix problems Quantum control. For a given a subgroup K of the group of unitary matrices and for given density matrices A and B, determine min UAU B F = A 2 F + B 2 F 2 max Re tr (UAU B ). U K U K

38 Some sample matrix problems Quantum control. For a given a subgroup K of the group of unitary matrices and for given density matrices A and B, determine min UAU B F = A 2 F + B 2 F 2 max Re tr (UAU B ). U K U K Here X F = (tr X X) 1/2 is the Frobenius norm.

39 Some sample matrix problems Quantum control. For a given a subgroup K of the group of unitary matrices and for given density matrices A and B, determine min UAU B F = A 2 F + B 2 F 2 max Re tr (UAU B ). U K U K Here X F = (tr X X) 1/2 is the Frobenius norm. More generally, for given density matrices A 0, A 1,..., A m and t 1,..., t m 0 summing up to 1, determine m min t j U j A j Uj A 0 : U 1,..., U m K. F j=1

40 Construction of quantum operations Let A 1,..., A m M r and B 1,..., B m M s be density matrices.

41 Construction of quantum operations Let A 1,..., A m M r and B 1,..., B m M s be density matrices. Can we find a quantum operation (trace preserving completely positive linear map) Φ : M r M s such that Φ(A j ) = B j for j = 1,..., m?

42 Construction of quantum operations Let A 1,..., A m M r and B 1,..., B m M s be density matrices. Can we find a quantum operation (trace preserving completely positive linear map) Φ : M r M s such that Φ(A j ) = B j for j = 1,..., m? This can be viewed as an interpolating problem.

43 Quantum error correction For a given quantum channel Φ : M n M n, can we find a k-dimensional quantum error correction code,

44 Quantum error correction For a given quantum channel Φ : M n M n, can we find a k-dimensional quantum error correction code, i.e., a k-dimensional subspace V of C n such that Φ(A) = A for all A M n satisfying P V AP V = A, where P V is the orthogonal projection of C n onto V.

45 Quantum error correction For a given quantum channel Φ : M n M n, can we find a k-dimensional quantum error correction code, i.e., a k-dimensional subspace V of C n such that Φ(A) = A for all A M n satisfying P V AP V = A, where P V is the orthogonal projection of C n onto V. This gives rise to problems in rank k-numerical ranges Λ k (A) C.

46 General questions and formalisms In general, given Hermitian matrices A 1,..., A m, determine submatrices of UA 1 U,..., UA m U with special structure.

47 General questions and formalisms In general, given Hermitian matrices A 1,..., A m, determine submatrices of UA 1 U,..., UA m U with special structure. One may use the theory in operator algebras to provide the formalisms.

48 General questions and formalisms In general, given Hermitian matrices A 1,..., A m, determine submatrices of UA 1 U,..., UA m U with special structure. One may use the theory in operator algebras to provide the formalisms. Researchers also used algebraic techniques, topological techniques, etc. to study the problems.

49 General questions and formalisms In general, given Hermitian matrices A 1,..., A m, determine submatrices of UA 1 U,..., UA m U with special structure. One may use the theory in operator algebras to provide the formalisms. Researchers also used algebraic techniques, topological techniques, etc. to study the problems. There are also problems on crytology, complexity, etc.

50 General questions and formalisms In general, given Hermitian matrices A 1,..., A m, determine submatrices of UA 1 U,..., UA m U with special structure. One may use the theory in operator algebras to provide the formalisms. Researchers also used algebraic techniques, topological techniques, etc. to study the problems. There are also problems on crytology, complexity, etc. Conclusion It is a wonderful interdisciplinary rsearch area. You are welcome to explore more and join the club!