Quantum Circuit Synthesis Roel Van Beeumen rvanbeeumen@lbl.gov - - PowerPoint PPT Presentation

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Quantum Circuit Synthesis Roel Van Beeumen rvanbeeumen@lbl.gov www.roelvanbeeumen.be CSSS Talk June 16, 2020 Outline 1 From Eigenvalues to Quantum Computing 2 Qubits and Quantum Circuits 3 Quantum Fourier Transform R. Van Beeumen (CRD)

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Quantum Circuit Synthesis

Roel Van Beeumen

rvanbeeumen@lbl.gov

www.roelvanbeeumen.be

CSSS Talk June 16, 2020

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Outline

1 From Eigenvalues to Quantum Computing 2 Qubits and Quantum Circuits 3 Quantum Fourier Transform

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 1

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1987 2005 2010 2015 2020

  • Belgium
  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 2

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Belgium

Capital: Brussels Population: 11,000,000 King: Filip I

  • R. Van Beeumen (CRD)

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1987 2005 2010 2015 2020

  • Belgium

BS Eng. MS Eng. PhD

STEM Career: 2005–2008: BS Mechanical-Electrical Engineering 2008–2010: MS Mathematical Engineering 2011–2015: PhD Computer Science KU Leuven, University of Leuven, Belgium

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 4

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KU Leuven, University of Leuven

Founded: 1425 Students: 50,000 Tuition: $1,000

  • R. Van Beeumen (CRD)

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1987 2005 2010 2015 2020

  • Belgium

BS Eng. MS Eng. PhD

PhD Thesis:

Rational Krylov methods for nonlinear eigenvalue problems

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 6

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My Research

b

a

f (x)dx Mathematics Engineering Informatics

  • R. Van Beeumen (CRD)

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Eigenvalue problems

  • R. Van Beeumen (CRD)

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Eigenvalue problems

  • R. Van Beeumen (CRD)

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Linear eigenvalue problem

x1 x2 xn

The eigenvalues and eigenmodes of a string are the solution of      a11 a12 . . . a1n a21 a22 . . . a1n . . . . . . ... . . . an1 an2 . . . ann     

  • A

     x1 x2 . . . xn     

x

= λ      x1 x2 . . . xn     

x

where λ is an eigenvalue x is an eigenvector

  • R. Van Beeumen (CRD)

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Quadratic eigenvalue problem

Vibration analysis in structural analysis gives rise to (λ2M + λC + K)x = 0 where λ is an eigenvalue x is an eigenvector M is the mass matrix C is the damping matrix K is the stiffness matrix

  • R. Van Beeumen (CRD)

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Nonlinear damping

Clamped beam:

  • λ2M + λC + K
  • x = 0

|C|

|λ| Clamped sandwich beam:

  • λ2M + C(λ) + K
  • x = 0

|λ|

|C(λ)|

for λ on the imaginary axis

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 12

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Active damping

Active damping in cars:

input

  • utput

System Controller

Delay eigenvalue problem

  • λ2M + λC + K + e−λτE
  • x = 0
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1987 2005 2010 2015 2020

  • Belgium

BS Eng. MS Eng. PhD BA Arch. MA Arch.

STEM Career: 2005–2008: BS Mechanical-Electrical Engineering 2006–2010: BA Archaeology 2008–2010: MS Mathematical Engineering 2010–2011: MA Archaeology 2011–2015: PhD Computer Science KU Leuven, University of Leuven, Belgium

  • R. Van Beeumen (CRD)

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Sagalassos Archaeological Research Project (Turkey)

  • R. Van Beeumen (CRD)

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Sagalassos Archaeological Research Project (Turkey)

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 16

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1987 2005 2010 2015 2020

  • Belgium

BS Eng. MS Eng. PhD BA Arch. MA Arch. Postdoc 1 Postdoc 2

STEM Career: 2005–2008: BS Mechanical-Electrical Engineering 2006–2010: BA Archaeology 2008–2010: MS Mathematical Engineering 2010–2011: MA Archaeology 2011–2015: PhD Computer Science 2015–2016: Postdoc @ KU Leuven 2016–2019: Postdoc @ Berkeley Lab

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Postdoc @ Berkeley Lab

LBNL Postdoc:

→ Computing Sciences Area → Computational Research Division → Applied Mathematics Department → Scalable Solvers Group Research Projects: Eigenvalue problems Model order reduction Numerical software development

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1987 2005 2010 2015 2020

  • Belgium

BS Eng. MS Eng. PhD BA Arch. MA Arch. Postdoc 1 Postdoc 2 Scientist

Since 2019: Career-track Research Scientist @ Berkeley Lab 2019 LDRD Early Career Award

Project: Approximate Unitary Matrix Decompositions for Quantum Circuit Synthesis 1st Postdoc: Daan Camps

  • R. Van Beeumen (CRD)

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Compiling Quantum Programs: Quantum Circuit Synthesis

Quantum Applications Quantum Chip Quantum Program

U

n qubits n qubits

encode 2n states encode 2n states

a quantum program U, is a unitary matrix of size 2n × 2n, too large to write down for large n

− →

Quantum Circuit

u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12

  • is a series of quantum gates,

each performing a simple unitary transformation on only a few qubits

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Qubits and Quantum Circuits

  • R. Van Beeumen (CRD)

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Classical Bit versus Qubit

Classical bit 2 states: 0 and 1 Quantum bit linear combinations: |ψ = α|0 + β|1 Computational basis states |0 := 1

  • |1 :=

1

  • |α|2 + |β|2 = 1
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Kronecker Products

Kronecker product of matrices A and B A ⊗ B :=      a11B a12B · · · a1,mB a21B a22B · · · a2,mB . . . . . . ... . . . an,1B an,2B · · · an,mB      Properties (γA) ⊗ B = A ⊗ (γB) = γ(A ⊗ B) A ⊗ (B + C) = A ⊗ B + A ⊗ C (B + C) ⊗ A = B ⊗ A + C ⊗ A and (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)

  • R. Van Beeumen (CRD)

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Unit Vectors and Identity Matrix

Unit vectors e1 = 1

  • = |0

e2 = 1

  • = |1

Identity matrix I2 = 1 1

  • =

1

  • E1=e1e⊤

1

+ 1

  • E2=e2e⊤

2

Direct sum A ⊕ B = A B

  • = E1 ⊗ A + E2 ⊗ B
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Multiple qubits

2 qubits: |ψ = α |00 + β |01 + γ |10 + δ |11

|00 :=     1     |01 :=     1     |10 :=     1     |11 :=     1     |α|2 + |β|2 + |γ|2 + |δ|2 = 1

n qubits: state space of dimension 2n linear combination of 2n computational basis states |ψ =

2

  • j1,...,jn=1

αj1j2···jn

  • ej1 ⊗ ej2 ⊗ · · · ⊗ ejn
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Quantum Circuits

Matrix notation φ = Uψ Quantum circuit |ψ U |φ φ = Um · · · U2U1ψ |ψ U1 U2 · · · Um |φ U ⊗ I U Uctr = I ⊕ U = I U

  • U
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Quantum Gates

Hadamard H 1 √ 2 1 1 1 −1

  • Pauli-X

X 1 1

  • Phase

S 1 i

  • Pauli-Y

Y −i i

  • π/8

T 1 eiπ/4

  • Pauli-Z

Z 1 −1

  • Controlled-NOT (CNOT)

   1 1 1 1    

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CNOT gate: definition

1st qubit q1: control 2nd qubit q2: target q1

  • q2

= X controlled operation: Uctr = I ⊕ X = E1 ⊗ I + E2 ⊗ X controlled NOT: CNOT = 1

  • E1

⊗ 1 1

  • I

+ 1

  • E2

⊗ 1 1

  • X

=     1 1 1 1    

  • R. Van Beeumen (CRD)

Quantum Fourier Transform June 16, 2020 28

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CNOT gate: behavior

target bit = |0 |0

  • |0

|φ |φ   

1 1 1 1

     

∗ ∗

   =   

∗ ∗

   = |0φ target bit = |1 |1

  • |1

|0 |1   

1 1 1 1

     

1

   =   

1

   = |11 |1

  • |1

|1 |0   

1 1 1 1

     

1

   =   

1

   = |10

|00 :=     1     |01 :=     1     |10 :=     1     |11 :=     1    

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Quantum Fourier Transform June 16, 2020 29

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SWAP gate: implemented by 3 CNOTs

SWAP gate: q1 × q2 q2 × q1 can be implemented by 3 CNOTs ×

  • ×

=

  • =

    1 1 1 1    

|00 :=     1     |01 :=     1     |10 :=     1     |11 :=     1    

  • R. Van Beeumen (CRD)

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Quantum Fourier Transform

  • R. Van Beeumen (CRD)

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Discrete Fourier Transform (DFT)

x: vector of length N − → y: vector of length N yk = 1 √ N

N−1

  • j=0

ωkj

N xj,

with ωN := e

−2πi N

Matrix notation

y = FNx, FN := 1 √ N       

ω0

N

ω0

N

ω0

N

· · · ω0

N

ω0

N

ω1

N

ω2

N

· · · ωN−1

N

ω0

N

ω2

N

ω4

N

· · · ω2(N−1)

N

. . . . . . . . . ... . . . ω0

N

ωN−1

N

ω2(N−1)

N

· · · ω(N−1)(N−1)

N

      

Examples

F1 =

  • 1
  • ,

F2 = 1 √ 2 1 1 1 −1

  • ,

F3 = 1 √ 3   1 1 1 1 ω3 ω2

3

1 ω2

3

ω3   , F4 = 1 2     1 1 1 1 1 −i −1 i 1 −1 1 −1 1 i −1 −i    

Complexity Matvec: O

  • N2

− → FFT: O

  • N log(N)
  • −

→ QFT: O

  • (log(N)2
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Fast Fourier Transform (FFT)

applies Discrete Fourier Transform in O(N log N) by recursively using radix-2 decomposition of permuted DFT matrix (FN = PF ′

N)

F ′

N =

1 √ 2

  • F ′

N/2

F ′

N/2

F ′

N/2ΩN/2

−F ′

N/2ΩN/2

  • =

1 √ 2

  • F ′

N/2

F ′

N/2

IN/2 IN/2 ΩN/2 −ΩN/2

  • let N = 2n or n = log2(N)

FN = PM1M2 · · · Mn where Mk = I2n−k ⊗ 1 √ 2 I2k−1 I2k−1 Ω2k−1 −Ω2k−1

  • ,

k = 1, 2, . . . , n

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Quantum Fourier Transform June 16, 2020 33

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Quantum Fourier Transform (QFT)

recall the radix-2 decomposition (F ′

2n = F′ n)

F′

n =

F′

n−1

F′

n−1

In−1 Ωn−1 1 √ 2 In−1 In−1 In−1 −In−1

  • = (I2 ⊗ F′

n−1)(In−1 ⊕ Ωn−1)

  • Dn

(H ⊗ In−1) where Ωn = R2 ⊗ R3 ⊗ · · · ⊗ Rn ⊗ Rn+1, Rk := 1 ω2k

  • quantum circuit for diagonal blocks

q1 Dn

  • · · ·
  • q2

Ωn−1 · · · R2 . . . = = . . . qn−1 Rn−1 · · · qn Rn · · ·

  • R. Van Beeumen (CRD)

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QFT Circuit

q1 H Dn · · · Pn q2 H Dn−1 · · · . . . . . . qn−1 · · · H D2 qn · · · H complexity: O

  • (log(N)2

in matrix notation Fn = PnF′

n = PnM1M2 · · · Mn

where Mk = In−k ⊗ [Dk(H ⊗ Ik−1)] , k = 1, 2, . . . , n

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Conclusions

Deriving the QFT from the FFT

by decomposing diagonal factors in FFT decomposition extend radix-2 to radix-d QFT decomposition QFT decomposition and corresponding circuit is not unique

Reference: Camps, Van Beeumen, Yang. “Quantum Fourier Transform Revisited.” arXiv:2003.03011 Acknowledgments: LBNL LDRD Program under U.S. Department of Energy Contract No. DE-AC02-05CH11231

  • R. Van Beeumen (CRD)

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