Linear Algebra and Quantum Computing - ALA 2010 …ala2010.pmf.uns.ac.rs/presentations/3w0900cl.pdf · Linear Algebra and Quantum Computing Chi-Kwong Li Department of Mathematics

Linear Algebra and Quantum Computing

Chi-Kwong LiDepartment of Mathematics

The College of William and MaryWilliamsburg, Virginia, USA

Joint work with Yiu-Tung Poon (Iowa State University).

Chi-Kwong Li Linear Algebra Quantum Computing

General computing models

Input −→ Computing Unit −→ Output

Classical computing (Abacus)

Hardware - Beads and bars.

Input - Using finger skill to change the states of the device.

Processor - Mechanical process with algorithms based on elementaryarithmetic rules.

Output - Beads and bars, then recorded by brush and ink.










































Modern Computing (Digital Computer)

Hardware - Mechanical/electronic/transistors.

Input - Punch cards, keyboards, scanners, sounds, etc. all convertedto binary bits - (0, 1) sequences.

Processor - Manipulations of (0, 1) sequences using Boolean logic.

0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1

Output - (0, 1) sequences realized as visual images, which can beviewed or printed.






0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1







0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1







0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1







0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1







0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1



Quantum computing

−→ Quantum Computing UnitOptical lattices, NMR −→

Hardware - Super conductor, trapped ions, optical lattices, quantumdot, MNR, etc.

Input - Quantum states in a specific form - Quantum bits (Qubits).

Processor - Provide suitable environment for the quantum system ofqubits to evolve.

Output - Measurement of the resulting quantum states.

All these require the understanding of mathematics, physics,chemistry, computer sciences, engineering, etc.


Quantum computing








Quantum computing








Quantum computing








Quantum computing








Quantum computing








Mathematical formulation (by von Neumann)

Suppose a quantum system havetwo (discrete) measurable physicalsates, say, up spin and down spinof a particle represented by

| ↑〉 =

(10

)and | ↓〉 =

(01

).

Before measurement, the vector state may be in superposition staterepresented by a complex vector

v = |ψ〉 = α| ↑〉+ β| ↓〉 =

(αβ

)∈ C2, |α|2 + |β|2 = 1.

One can apply a quantum operation to a state in superposition.




| ↑〉 =

(10

)and | ↓〉 =

(01

).


v = |ψ〉 = α| ↑〉+ β| ↓〉 =

(αβ

)∈ C2, |α|2 + |β|2 = 1.





| ↑〉 =

(10

)and | ↓〉 =

(01

).


v = |ψ〉 = α| ↑〉+ β| ↓〉 =

(αβ

)∈ C2, |α|2 + |β|2 = 1.





| ↑〉 =

(10

)and | ↓〉 =

(01

).


v = |ψ〉 = α| ↑〉+ β| ↓〉 =

(αβ

)∈ C2, |α|2 + |β|2 = 1.



Matrix and Bloch sphere

It is convenient to represent thequantum state |ψ〉 as a rank-oneorthogonal projection:

|ψ〉〈ψ| = 12

(1 + z x− iyx+ iy 1− z

)with x, y, z ∈ R, x2 + y2 + z2 = 1.

Bloch sphere

States of k qubits are represented as vector in ⊗kC2 = C2k

, or2k × 2k density matrices.

One can apply a single quantum operation to ALL the states|x1 · · ·xk〉 simultaneously if

|ψ〉 =∑

xi∈{|↑〉,|↓〉}

γx1···k|x1〉 ⊗ · · · ⊗ |xk〉.




|ψ〉〈ψ| = 12

(1 + z x− iyx+ iy 1− z

)with x, y, z ∈ R, x2 + y2 + z2 = 1.

Bloch sphere




|ψ〉 =∑

xi∈{|↑〉,|↓〉}

γx1···k|x1〉 ⊗ · · · ⊗ |xk〉.




|ψ〉〈ψ| = 12

(1 + z x− iyx+ iy 1− z

)with x, y, z ∈ R, x2 + y2 + z2 = 1.

Bloch sphere




|ψ〉 =∑

xi∈{|↑〉,|↓〉}

γx1···k|x1〉 ⊗ · · · ⊗ |xk〉.


Mathematical tools

As a consequence of the Schrödinger equation

all quantum gates and quantum evolutions (for a closed system) areunitary similarity transforms of the density matrices representing thestates, i.e.,

ρ(t) 7→ U(t)ρ(0)U(t)∗ for some unitaries U(t).

By the results of Choi in 70’s and Kraus in 80’sQuantum channels, quantum operations, quantummeasurement operators, etc. aretrace preserving completely positivelinear maps of the form

ρ 7→∑r

j=1 FjρF∗j .

The theory was discovered waybefore the applications!

Man-Duen Choi


Mathematical tools

As a consequence of the Schrödinger equationall quantum gates and quantum evolutions (for a closed system) are

unitary similarity transforms of the density matrices representing thestates, i.e.,



ρ 7→∑r

j=1 FjρF∗j .


Man-Duen Choi


Mathematical tools

As a consequence of the Schrödinger equationall quantum gates and quantum evolutions (for a closed system) areunitary similarity transforms of the density matrices representing thestates, i.e.,



ρ 7→∑r

j=1 FjρF∗j .


Man-Duen Choi


Mathematical tools



By the results of Choi in 70’s and Kraus in 80’sQuantum channels, quantum operations, quantummeasurement operators, etc. are

trace preserving completely positivelinear maps of the form

ρ 7→∑r

j=1 FjρF∗j .


Man-Duen Choi


Mathematical tools




ρ 7→∑r

j=1 FjρF∗j .


Man-Duen Choi


Mathematical tools




ρ 7→∑r

j=1 FjρF∗j .


Man-Duen Choi


Linear Algebra

Let Mn be the set of n× n complex matrices,

Hn be the set of n× n complex Hermitian matrices.

A map L : Mn →Mm is completely positive if L admits an operator sumrepresentation

L(A) =r∑

j=1

FjAF∗j ,

where F1, . . . , Fr are m× n complex matrices.

In addition, L is unital if L(In) = Im, equivalently,∑r

j=1 FjF∗j = Im;

L is trace preserving if tr A = trL(A), equivalently,∑r

j=1 F∗j Fj = In.


Linear Algebra




L(A) =r∑

j=1

FjAF∗j ,



j=1 FjF∗j = Im;


j=1 F∗j Fj = In.


Linear Algebra




L(A) =r∑

j=1

FjAF∗j ,



j=1 FjF∗j = Im;


j=1 F∗j Fj = In.


Linear Algebra




L(A) =r∑

j=1

FjAF∗j ,



j=1 FjF∗j = Im;


j=1 F∗j Fj = In.


Linear Algebra




L(A) =r∑

j=1

FjAF∗j ,



j=1 FjF∗j = Im;


j=1 F∗j Fj = In.


A general problem

Since every quantum operation / channel is a trace preserving completelypositive linear map, it is interesting to study the following.

QuestionGiven A1, . . . , Ak ∈Mn and B1, . . . , Bk ∈Mm, is there a (unital/tracepreserving) completely positive linear map L satisfying

L(Aj) = Bj for all j = 1, . . . , k?

It is also (more?) interesting to consider the following related problems.

Determine / deduce properties of L based on the information ofL(A1), . . . ,L(Ak) for some special matrices A1, . . . , Ak.

Understand the duality relation between the trace preservingcompletely positive linear maps and the unital preserving completelypositive linear maps.


A general problem



L(Aj) = Bj for all j = 1, . . . , k?





A general problem



L(Aj) = Bj for all j = 1, . . . , k?





A general problem



L(Aj) = Bj for all j = 1, . . . , k?





Basic results

For A ∈ Hn with eigenvalues λ1(A) ≥ · · · ≥ λn(A), let

λ(A) = (λ1(A), . . . , λn(A)).

Definition

For x, y ∈ R1×m, we say that x is majorized by y, denoted by x ≺ y, ifthe sum of entries of x is the same as that of y, and the sum of the klargest entries of x is not larger than that of y for k = 1, . . . , k − 1.

Examples

(5, 4, 1) ≺ (7, 3, 0) (5, 4, 1) 6≺ (6, 2, 2), and (6, 2, 2) 6≺ (5, 4, 1).

It is known that x ≺ y if and only if there is a doubly stochastic matrixD such that x = yD.


Basic results


λ(A) = (λ1(A), . . . , λn(A)).

Definition


Examples

(5, 4, 1) ≺ (7, 3, 0) (5, 4, 1) 6≺ (6, 2, 2), and (6, 2, 2) 6≺ (5, 4, 1).



Basic results


λ(A) = (λ1(A), . . . , λn(A)).

Definition


Examples

(5, 4, 1) ≺ (7, 3, 0)

(5, 4, 1) 6≺ (6, 2, 2), and (6, 2, 2) 6≺ (5, 4, 1).



Basic results


λ(A) = (λ1(A), . . . , λn(A)).

Definition


Examples

(5, 4, 1) ≺ (7, 3, 0) (5, 4, 1) 6≺ (6, 2, 2),

and (6, 2, 2) 6≺ (5, 4, 1).



Basic results


λ(A) = (λ1(A), . . . , λn(A)).

Definition


Examples

(5, 4, 1) ≺ (7, 3, 0) (5, 4, 1) 6≺ (6, 2, 2), and (6, 2, 2) 6≺ (5, 4, 1).



Basic results


λ(A) = (λ1(A), . . . , λn(A)).

Definition


Examples

(5, 4, 1) ≺ (7, 3, 0) (5, 4, 1) 6≺ (6, 2, 2), and (6, 2, 2) 6≺ (5, 4, 1).



TheoremSuppose A ∈ Hn and B ∈ Hm. Let a+ (respectively, a−) be the sum ofthe positive (respectively, negative) eigenvalues of A.

The followingconditions are equivalent.

There is a trace preserving completely positive linear mapL : Mn →Mm such that L(A) = B.

λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.

There is an n×m row stochastic matrix (nonnegative matrix withall row sums equal to one) D such that λ(B) = λ(A)D.

The matrix D can be chosen so that the first k rows all equal andthe last n− k rows all equal.

One can use D to construct m× n matrices F1, . . . , Fr withr = max(m,n) such that

B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.

For density matrices A and B, the condition trivially holds.


TheoremSuppose A ∈ Hn and B ∈ Hm. Let a+ (respectively, a−) be the sum ofthe positive (respectively, negative) eigenvalues of A. The followingconditions are equivalent.


λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.





λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.





λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.





λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.





λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.





λ(B) ≺ (a+, 0, . . . , 0, a−) in R1×m.




B =r∑

j=1

FjAF∗j and

r∑j=1

F ∗j Fj = In.

For density matrices A and B, the condition trivially holds.Chi-Kwong Li Linear Algebra Quantum Computing

TheoremLet A ∈ Hn and B ∈ Hm. The following conditions are equivalent.

There is a unital completely positive linear map L such thatL(A) = B.

λn(A) ≤ λj(B) ≤ λ1(A) for all j = 1, . . . ,m.

There is an n×m column stochastic matrix D such thatλ(B) = λ(A)D.

Remark The condition may fail even if A and B are density matrices.

QuestionCan we deduce this result from the previous one using duality ofcompletely positive linear map?





































QuestionAssume there is a unital completely positive map sending A to B, andalso a trace preserving completely positive map sending A to B.

Is therea unital trace preserving completely positive map sending A to B?

The following example shows that the answer is negative.

Example

Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is atrace preserving completely positive map sending A to B, and also aunital completely positive map sending A to B. But there is no tracepreserving completely positive linear map sending A to B.Reason: If there were such a map, it would send A1 to B1 for

A1 = A− I4 = diag (3, 0, 0,−1) and B1 = B − I4 = diag (2, 2,−1,−1),

which is a contradiction.


QuestionAssume there is a unital completely positive map sending A to B, andalso a trace preserving completely positive map sending A to B. Is therea unital trace preserving completely positive map sending A to B?


Example







Example







Example

Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is atrace preserving completely positive map sending A to B,

and also aunital completely positive map sending A to B. But there is no tracepreserving completely positive linear map sending A to B.Reason: If there were such a map, it would send A1 to B1 for






Example

Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is atrace preserving completely positive map sending A to B, and also aunital completely positive map sending A to B.

But there is no tracepreserving completely positive linear map sending A to B.Reason: If there were such a map, it would send A1 to B1 for






Example

Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is atrace preserving completely positive map sending A to B, and also aunital completely positive map sending A to B. But there is no tracepreserving completely positive linear map sending A to B.

Reason: If there were such a map, it would send A1 to B1 for






Example





TheoremLet A,B ∈ Hn. The following conditions are equivalent.

There exists a unital trace preserving completely positive map Lsuch that L(A) = B.

For each t ∈ R, there exists a trace preserving completely positivemap L such that L(A− tI) = B − tI.

λ(B) ≺ λ(A). i.e., there is a doubly stochastic matrix D such thatλ(B) = λ(A)D.

There is a unitary U ∈Mn such that UAU∗ has diagonal entriesλ1(B), . . . , λn(B).

There exist unitary matrices Uj , 1 ≤ j ≤ n such that

B = 1n

∑nj=1 UjAU

∗j .

B is in the convex hull of the unitary orbit U(A) of A:

{UAU∗ : U unitary}.








B = 1n

∑nj=1 UjAU

∗j .










B = 1n

∑nj=1 UjAU

∗j .










B = 1n

∑nj=1 UjAU

∗j .










B = 1n

∑nj=1 UjAU

∗j .










B = 1n

∑nj=1 UjAU

∗j .










B = 1n

∑nj=1 UjAU

∗j .




Results and questions on multiple matrices

TheoremSuppose A1, . . . , Ak ∈Mn and B1, . . . , Bk ∈Mm are diagonal matrices.

Let d(R) is the vector of diagonal entries of the square matrix R. Thenthere is a unital / trace preserving / unital and trace preservingcompletely positive linear maps L such that

L(Aj) = Bj for j = 1, . . . , k

if and only if there is an n×m column / row / doubly stochastic matrixD such that d(Bj) = d(Aj)D for j = 1, . . . , k.

Evidently, the result can be applied to commuting families{A1, . . . , Ak} and {B1, . . . , Bk}.

The problem reduces to joint/multivariate majorization.

What about non-commuting families?



TheoremSuppose A1, . . . , Ak ∈Mn and B1, . . . , Bk ∈Mm are diagonal matrices.Let d(R) is the vector of diagonal entries of the square matrix R.

Thenthere is a unital / trace preserving / unital and trace preservingcompletely positive linear maps L such that

L(Aj) = Bj for j = 1, . . . , k







TheoremSuppose A1, . . . , Ak ∈Mn and B1, . . . , Bk ∈Mm are diagonal matrices.Let d(R) is the vector of diagonal entries of the square matrix R. Thenthere is a unital / trace preserving / unital and trace preservingcompletely positive linear maps L such that

L(Aj) = Bj for j = 1, . . . , k








L(Aj) = Bj for j = 1, . . . , k








L(Aj) = Bj for j = 1, . . . , k








L(Aj) = Bj for j = 1, . . . , k








L(Aj) = Bj for j = 1, . . . , k






CP maps with restricted Kraus (Choi) rank

For given A ∈ Hn, B ∈ Hm, and a postive integer r, we are interested inconstructing/finding a CP map L : Mn →Mm of the formL(A) =

∑rj=1 FjAF

∗j such that L(A) = B.

TheoremLet A ∈ Hn and B ∈ Hm. Suppose nr ≥ m. The following conditionsare equivalent.

There exist m×n matrices F1, . . . , Fr such that∑r

j=1 FjAF∗j = B.

There is an m× (nr) matrix F such that F (A⊗ Ir)F∗.

The number of positive (negative) eigenvalues of A⊗ Ir is morethan or equal to the number of positive (negative) eigenvalues of B.




∑rj=1 FjAF




j=1 FjAF∗j = B.






∑rj=1 FjAF




j=1 FjAF∗j = B.






∑rj=1 FjAF




j=1 FjAF∗j = B.





There exist m× n matrices F1, . . . , Fr such that∑r

j=1 FjF∗j = Im

and∑r

j=1 FjAF∗j = B.

B is a compression of A⊗ Ir, i.e. B = F (A⊗ Ir)F∗ for some

m× (nr) F such that FF ∗ = Im.

The eigenvalues of A⊗ Ir interlace those of B, i.e.,

λj(A) ≥ λ(j−1)r+1(B) whenever (j − 1)r + 1 ≤ m, and

λn−j+1(A) ≤ bm−(j−1)r(B) whenever 1 ≤ m− (j − 1)r.

Remark Even if A,B ∈ Hn are density matrices, the roles of A and Bare not symmetric in the last two theorems.




j=1 FjF∗j = Im

and∑r

j=1 FjAF∗j = B.










j=1 FjF∗j = Im

and∑r

j=1 FjAF∗j = B.










j=1 FjF∗j = Im

and∑r

j=1 FjAF∗j = B.










j=1 FjF∗j = Im

and∑r

j=1 FjAF∗j = B.








TheoremLet A ∈ Hn and B ∈ Hm. Suppose mr ≥ n. The following conditionsare equivalent.


j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.

The matrix A⊕ 0mr−n is unitarily similar to (Cij) such thatC11, . . . , Crr ∈ Hm such that B = C11 + · · ·+ Crr.

There exist C1, . . . , Cr ∈ Hm such that B = C1 + · · ·+ Cr and(A+ µI)⊕ µIrm−n = C̃1 + · · ·+ C̃r, where µ = max{−λn(A), 0}C̃j is unitarily similar to (Cj + µI)⊕ 0(r−1)m.

Remark Conditions (b) and (c) can be expressed in terms of eigenvalueinequalities using Littlewood-Richardson rules or the Horn sequences:∑

j∈J0

λj(B) ≤∑j∈J1

λj(C1) + · · ·+∑j∈Jr

λj(Cr)

with (J0, J1, . . . , Jr) ∈ S(n, r, `) for ` ∈ {1, . . . , n− 1}.




j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.




j∈J0

λj(B) ≤∑j∈J1

λj(C1) + · · ·+∑j∈Jr

λj(Cr)





j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.




j∈J0

λj(B) ≤∑j∈J1

λj(C1) + · · ·+∑j∈Jr

λj(Cr)





j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.




j∈J0

λj(B) ≤∑j∈J1

λj(C1) + · · ·+∑j∈Jr

λj(Cr)





j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.




j∈J0

λj(B) ≤∑j∈J1

λj(C1) + · · ·+∑j∈Jr

λj(Cr)



CorollaryLet A,B ∈ Hn be positive semidefinite. The following conditions areequivalent.

There exist n× n matrices F1, . . . , Fr such that∑r

j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.

There exist C1, . . . , Cr ∈ Hn and unitary U1, . . . , Ur ∈Mn suchthat A =

∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .

There exist n× n matrices F̃1, . . . , F̃r such that∑r

j=1 F̃∗j F̃j = In

and∑r

j=1 F̃jBF̃∗j = A.

Corollary

Let A,B ∈ Hn be positive semidefinite. Then B = L(A) for some TPCPmap L of with rank r if and only if A = L̃(B) for some TPCP map L̃with rank r.

Remark This is not true for unital completely positive linear maps.




j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.


∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .



and∑r


Corollary






j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.


∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .



and∑r


Corollary






j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.


∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .



and∑r


Corollary






j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.


∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .



and∑r


Corollary






j=1 F∗j Fj = In

and∑r

j=1 FjAF∗j = B.


∑rj=1 U

∗j CjUj and B =

∑rj=1 Cj .



and∑r


Corollary




A computaional approach

Derive numerical scheme (using gradient flow, positive semi-definiteprogramming, etc.) to solve the following:Given A1, . . . , Ak ∈ Hn, B1, . . . , Bk ∈ Hm, determine L such that

L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.

We may impose additional conditons such as:

L(In) = Im (unital).

trL(Eij) = δij for 1 ≤ i, j ≤ n (trace preserving).

The sum of r × r principal submatrix of L: Sr(L) = 0 for a given r(L has rank less than r).




L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.








L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.








L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.








L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.








L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.








L(Aj) = Bj , j = 1, . . . , k,

and[L(Eij)] ≥ 0.






Conclusion

Quantum computing is a fascinating subject with many interestingproblems in linear algebra.

Many techniques such as unitary orbits, completely positive linearmaps, majorization, dilation theory, spectral inequalities, positivesemi-definite programming, optimization, etc. are useful.

There are much opportunities for further research.

One may prove results to show that quantum computing arerealistic / impossible!

Welcome to join the club!


Conclusion







Conclusion







Conclusion







Conclusion







Conclusion







Thank you for your attention!


Thank you for your attention!


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Linear Algebra and Quantum Computing - ALA 2010 …ala2010.pmf.uns.ac.rs/presentations/3w0900cl.pdf · Linear Algebra and Quantum Computing Chi-Kwong Li Department of Mathematics