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) Quantum Fourier Transform June 16, 2020 1
◦ Belgium 1987 2005 2010 2015 2020 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 2
Belgium Capital: Brussels Population: 11,000,000 King: Filip I R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 3
◦ Belgium BS Eng. MS Eng. PhD 1987 2005 2010 2015 2020 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
KU Leuven, University of Leuven Founded: 1425 Students: 50,000 Tuition: $1,000 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 5
◦ Belgium BS Eng. MS Eng. PhD 1987 2005 2010 2015 2020 PhD Thesis: Rational Krylov methods for nonlinear eigenvalue problems R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 6
My Research Engineering Mathematics � b f ( x ) dx a Informatics R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 7
Eigenvalue problems R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 8
Eigenvalue problems R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 9
Linear eigenvalue problem x 2 x 1 x n The eigenvalues and eigenmodes of a string are the solution of a 11 a 12 . . . a 1 n x 1 x 1 a 21 a 22 . . . a 1 n x 2 x 2 = λ . . . . . ... . . . . . . . . . . . . . a n 1 a n 2 a nn x n x n � �� � � �� � � �� � x x A where λ is an eigenvalue x is an eigenvector R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 10
Quadratic eigenvalue problem Vibration analysis in structural analysis gives rise to ( λ 2 M + λ C + 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) Quantum Fourier Transform June 16, 2020 11
Nonlinear damping Clamped sandwich beam: Clamped beam: � � � � λ 2 M + λ C + K λ 2 M + C ( λ ) + K x = 0 x = 0 | C | | C ( λ ) | | λ | | λ | for λ on the imaginary axis R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 12
Active damping Active damping in cars: input output System Controller Delay eigenvalue problem � � λ 2 M + λ C + K + e − λτ E x = 0 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 13
◦ Belgium BS Eng. MS Eng. PhD 1987 2005 BA Arch. 2010 MA Arch. 2015 2020 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) Quantum Fourier Transform June 16, 2020 14
Sagalassos Archaeological Research Project (Turkey) R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 15
Sagalassos Archaeological Research Project (Turkey) R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 16
◦ Belgium BS Eng. MS Eng. PhD Postdoc 1 Postdoc 2 1987 2005 BA Arch. 2010 MA Arch. 2015 2020 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 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 17
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 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 18
◦ Belgium BS Eng. MS Eng. PhD Postdoc 1 Postdoc 2 Scientist 1987 2005 BA Arch. 2010 MA Arch. 2015 2020 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) Quantum Fourier Transform June 16, 2020 19
Compiling Quantum Programs: Quantum Circuit Synthesis Quantum Applications Quantum Chip Quantum Program Quantum Circuit u 1 • u 3 • u 2 u 4 u 6 u 8 u 10 • • n qubits U n qubits − → u 5 u 7 • • encode 2 n states encode 2 n states u 9 u 12 • • u 11 a quantum program U , is a is a series of quantum gates, unitary matrix of size 2 n × 2 n , each performing a simple unitary too large to write down for large n transformation on only a few qubits R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 20
Qubits and Quantum Circuits R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 21
Classical Bit versus Qubit Classical bit Quantum bit 2 states: 0 and 1 linear combinations: | ψ � = α | 0 � + β | 1 � Computational basis states � 1 � � 0 � | α | 2 + | β | 2 = 1 | 0 � := | 1 � := 0 1 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 22
Kronecker Products Kronecker product of matrices A and B a 11 B a 12 B · · · a 1 , m B · · · a 21 B a 22 B a 2 , m B A ⊗ B := . . . ... . . . . . . a n , 1 B a n , 2 B · · · a n , m B Properties ( γ A ) ⊗ B = A ⊗ ( γ B ) = 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) Quantum Fourier Transform June 16, 2020 23
Unit Vectors and Identity Matrix Unit vectors � 1 � � 0 � e 1 = = | 0 � e 2 = = | 1 � 0 1 Identity matrix � 1 � � 1 � � 0 � 0 0 0 I 2 = = + 0 1 0 0 0 1 � �� � � �� � E 1 = e 1 e ⊤ E 2 = e 2 e ⊤ 1 2 Direct sum � A � A ⊕ B = = E 1 ⊗ A + E 2 ⊗ B B R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 24
Multiple qubits 2 qubits: | ψ � = α | 00 � + β | 01 � + γ | 10 � + δ | 11 � 1 0 0 0 0 1 0 0 | α | 2 + | β | 2 + | γ | 2 + | δ | 2 = 1 | 00 � := | 01 � := | 10 � := | 11 � := 0 0 1 0 0 0 0 1 n qubits: state space of dimension 2 n linear combination of 2 n computational basis states 2 � � � | ψ � = α j 1 j 2 ··· j n e j 1 ⊗ e j 2 ⊗ · · · ⊗ e j n j 1 ,..., j n =1 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 25
Quantum Circuits Quantum circuit Matrix notation | ψ � | φ � φ = U ψ U · · · | ψ � U 1 U 2 U m | φ � φ = U m · · · U 2 U 1 ψ U U ⊗ I � I � • U ctr = I ⊕ U = U U R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 26
Quantum Gates � 1 � � 0 � 1 1 1 √ Hadamard Pauli- X H X 1 − 1 1 0 2 � 1 � � 0 � 0 − i Phase Pauli- Y S Y 0 0 i i � 1 � � 1 � 0 0 π/ 8 Pauli- Z T Z e i π/ 4 0 0 − 1 1 • 1 Controlled-NOT (CNOT) 0 1 1 0 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 27
CNOT gate: definition q 1 • • 1st qubit q 1 : control = q 2 2nd qubit q 2 : target X controlled operation: U ctr = I ⊕ X = E 1 ⊗ I + E 2 ⊗ X controlled NOT: 1 � 1 � � 1 � � 0 � � 0 � 0 0 0 1 1 CNOT = ⊗ + ⊗ = 0 0 0 1 0 1 1 0 0 1 � �� � � �� � � �� � � �� � 1 0 E 1 I E 2 X R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 28
CNOT gate: behavior target bit = | 0 � 1 ∗ ∗ | 0 � • | 0 � 1 ∗ ∗ = = | 0 φ � | φ � | φ � 0 1 0 0 1 0 0 0 target bit = | 1 � 1 0 0 | 1 � • | 1 � 1 0 0 = = | 11 � 1 0 0 1 1 0 | 0 � | 1 � 0 1 | 00 � := | 01 � := 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 | 1 � • | 1 � | 10 � := | 11 � := 1 0 0 1 0 = = | 10 � 0 1 | 1 � | 0 � 0 1 0 1 1 0 1 0 R. Van Beeumen (CRD) Quantum Fourier Transform June 16, 2020 29
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