Download Quantum computing and mathematical research

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