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Quantum computing and mathematical research Chi-Kwong Li The College of William and Mary Chi-Kwong Li Quantum computing Classical computing Chi-Kwong Li Quantum computing Classical computing Hardware - Beads and bars. Chi-Kwong Li Quantum computing Classical computing Hardware - Beads and bars. Input - Using finger skill to change the states of the device. Chi-Kwong Li Quantum computing 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. Chi-Kwong Li Quantum computing 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. Chi-Kwong Li Quantum computing Modern computing Chi-Kwong Li Quantum computing Modern computing Hardware - Mechanical/electronic/integrated circuits. Chi-Kwong Li Quantum computing Modern computing Hardware - Mechanical/electronic/integrated circuits. Input - Punch cards, keyboards, scanners, sounds, etc. all converted to binary bits - (0, 1) sequences. Chi-Kwong Li Quantum computing 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. Chi-Kwong Li Quantum computing Modern computing 0∨0=0 0∨1=1 1∨0=1 1∨1=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. Chi-Kwong Li Quantum computing Modern computing 0∨0=0 0∨1=1 1∨0=1 1∨1=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. Chi-Kwong Li Quantum computing Quantum computing Chi-Kwong Li Quantum computing Quantum computing Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Chi-Kwong Li Quantum computing Quantum computing Hardware - Super conductor, trapped ions, optical lattices, quantum dot, MNR, etc. Input - Quantum states in a specific form - Qubit (Quantum bit). Chi-Kwong Li Quantum computing Quantum computing 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. Chi-Kwong Li Quantum computing Quantum computing 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. Chi-Kwong Li Quantum computing Quantum computing 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. Chi-Kwong Li Quantum computing Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment. Chi-Kwong Li Quantum computing Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment. Dr. Quantum - Entanglement. Chi-Kwong Li Quantum computing Some quantum physics background from Youtube! Dr. Quantum - Two-slit experiment. Dr. Quantum - Entanglement. Quantum teleportation. Chi-Kwong Li Quantum computing Mathematical challenges How to control the (initial) quantum states? Chi-Kwong Li Quantum computing Mathematical challenges How to control the (initial) quantum states? How to create the appropriate environment for the quantum mechanical system to evolve without observing? Chi-Kwong Li Quantum computing 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)? Chi-Kwong Li Quantum computing 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. Chi-Kwong Li Quantum computing Quantum states and quantum bits Suppose a quantum system have two (discrete) measurable physical sates, say, up spin and down spin of a particle. Chi-Kwong Li Quantum computing 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 1 0 |0i = and |1i = . 0 1 Chi-Kwong Li Quantum computing 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 1 0 |0i = and |1i = . 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 = |ψi = α|0i + β|1i = ∈ C2 , |α|2 + |β|2 = 1. β Chi-Kwong Li Quantum computing 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 1 0 |0i = and |1i = . 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 = |ψi = α|0i + β|1i = ∈ C2 , |α|2 + |β|2 = 1. β In quantum mechanics/computing, one uses such a basic quantum state/ quantum bit (qubit). Chi-Kwong Li Quantum computing In fact, it is more convenient to represent the quantum state |ψi as a rank one orthogonal projection: 1 1 + z x + iy Q = vv ∗ = |ψihψ| = 2 x − iy 1 − z with x, y, z ∈ R such that x2 + y 2 + z 2 = 1. Chi-Kwong Li Quantum computing In fact, it is more convenient to represent the quantum state |ψi as a rank one orthogonal projection: 1 1 + z x + iy Q = vv ∗ = |ψihψ| = 2 x − iy 1 − z with x, y, z ∈ R such that x2 + y 2 + z 2 = 1. There is a Bloch sphere representation of a qubit Chi-Kwong Li Quantum computing Quantum gates and quantum operations The state of k qubits is represented as the tensor product of k 2 × 2 matrices Q1 ⊗ · · · ⊗ Qk . Chi-Kwong Li Quantum computing Quantum gates and quantum operations The state of k qubits is represented as the tensor product of k 2 × 2 matrices Q1 ⊗ · · · ⊗ Qk . To accommodate various quantum effects, one considers 2k × 2k density matrices, i.e., trace one positive semidefinite Hermitian matrices. Chi-Kwong Li Quantum computing 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) 7→ U (t)A(0)U (t)∗ Chi-Kwong Li for unitaries U (t). Quantum computing 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) 7→ 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 Chi-Kwong Li Quantum computing 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) 7→ 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 7→ r X Xi AXj∗ , j=1 with Pr ∗ j=1 Xj Xj = I in case Φ is trace preserving. Chi-Kwong Li Quantum computing 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 kU AU ∗ −BkF = kAk2F +kBk2F −2 max Re tr (U AU ∗ B ∗ ). U ∈K U ∈K Chi-Kwong Li Quantum computing 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 kU AU ∗ −BkF = kAk2F +kBk2F −2 max Re tr (U AU ∗ B ∗ ). U ∈K U ∈K Here kXkF = (tr X ∗ X)1/2 is the Frobenius norm. Chi-Kwong Li Quantum computing 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 kU AU ∗ −BkF = kAk2F +kBk2F −2 max Re tr (U AU ∗ B ∗ ). U ∈K U ∈K Here kXkF = (tr X ∗ X)1/2 is the Frobenius norm. More generally, for given density matrices A0 , A1 , . . . , Am and t1 , . . . , tm ≥ 0 summing up to 1, determine m X ∗ min t U A U − A : U , . . . , U ∈ K . j j j j 0 1 m j=1 F Chi-Kwong Li Quantum computing Construction of quantum operations Let A1 , . . . , Am ∈ Mr and B1 , . . . , Bm ∈ Ms be density matrices. Chi-Kwong Li Quantum computing Construction of quantum operations Let A1 , . . . , Am ∈ Mr and B1 , . . . , Bm ∈ Ms be density matrices. Can we find a quantum operation (trace preserving completely positive linear map) Φ : Mr → Ms such that Φ(Aj ) = Bj Chi-Kwong Li for j = 1, . . . , m? Quantum computing Construction of quantum operations Let A1 , . . . , Am ∈ Mr and B1 , . . . , Bm ∈ Ms be density matrices. Can we find a quantum operation (trace preserving completely positive linear map) Φ : Mr → Ms such that Φ(Aj ) = Bj for j = 1, . . . , m? This can be viewed as an interpolating problem. Chi-Kwong Li Quantum computing Quantum error correction For a given quantum channel Φ : Mn → Mn , can we find a k-dimensional quantum error correction code, Chi-Kwong Li Quantum computing Quantum error correction For a given quantum channel Φ : Mn → Mn , can we find a k-dimensional quantum error correction code, i.e., a k-dimensional subspace V of Cn such that Φ(A) = A for all A ∈ Mn satisfying PV APV = A, where PV is the orthogonal projection of Cn onto V . Chi-Kwong Li Quantum computing Quantum error correction For a given quantum channel Φ : Mn → Mn , can we find a k-dimensional quantum error correction code, i.e., a k-dimensional subspace V of Cn such that Φ(A) = A for all A ∈ Mn satisfying PV APV = A, where PV is the orthogonal projection of Cn onto V . This gives rise to problems in rank k-numerical ranges Λk (A) ⊆ C. Chi-Kwong Li Quantum computing General questions and formalisms In general, given Hermitian matrices A1 , . . . , Am , determine submatrices of U A1 U ∗ , . . . , U Am U ∗ with special structure. Chi-Kwong Li Quantum computing General questions and formalisms In general, given Hermitian matrices A1 , . . . , Am , determine submatrices of U A1 U ∗ , . . . , U Am U ∗ with special structure. One may use the theory in operator algebras to provide the formalisms. Chi-Kwong Li Quantum computing General questions and formalisms In general, given Hermitian matrices A1 , . . . , Am , determine submatrices of U A1 U ∗ , . . . , U Am 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. Chi-Kwong Li Quantum computing General questions and formalisms In general, given Hermitian matrices A1 , . . . , Am , determine submatrices of U A1 U ∗ , . . . , U Am 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. Chi-Kwong Li Quantum computing General questions and formalisms In general, given Hermitian matrices A1 , . . . , Am , determine submatrices of U A1 U ∗ , . . . , U Am 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! Chi-Kwong Li Quantum computing