Foundations of Quantum Programming

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Foundations of Quantum Programming

Mingsheng Ying

University of Technology Sydney, Australia Institute of Software, Chinese Academy of Sciences

Tsinghua University, China

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Outline

Introduction

Syntax of Quantum Programs Operational Semantics Denotational Semantics Quantum Hoare Logic Research Problems

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Outline

Introduction

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How to program quantum computers?

I Quantum algorithms:

Deutsch-Josza, Grover, Shor, HHL, ...

Quantum algorithm zoo (http://math.nist.gov/quantum/zoo/)

I Quantum computers:

Google (53 qubits —quantum supremacy— 2019) IBM Q (50 qubits — 2018)

Intel, Rigetti, IonQ, ...

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How to program quantum computers?

I Quantum algorithms:

Deutsch-Josza, Grover, Shor, HHL, ...

Quantum algorithm zoo (http://math.nist.gov/quantum/zoo/)

I Quantum computers:

Google (53 qubits —quantum supremacy— 2019) IBM Q (50 qubits — 2018)

Intel, Rigetti, IonQ, ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I Quipper @Dalhousie

I isQ @Institute of Software, Chinese Academy of Sciences

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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Quantum programming platforms

I Q# @Microsoft

I Cirq @Google

I Qiskit @IBM

I Quil @Riggeti

I ...

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This lecture

I Principlesunderlyingallof the quantum programming languages

I Notthe languages themselves.

I Reference:

M. S. Ying, Foundations of Quantum Programming, Morgan Kaufmann 2016,Chapters 3 and 4.

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This lecture

I Reference:

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This lecture

I Reference:

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Programming languages and tools

Programming languages are notations used forspecifying,organising andreasoningabout computations.

[R. Sethi, Programming languages: Concepts and Constructs]

I Semantics

I Turing-complete?

I Compilers

I Program test, debugging, analysis

I How to verify correctness of your programs?

I ...

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Programming languages and tools

I Semantics

I Turing-complete?

I Compilers

I ...

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Programming languages and tools

I Semantics

I Turing-complete?

I Compilers

I ...

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Programming languages and tools

I Semantics

I Turing-complete?

I Compilers

I ...

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Programming languages and tools

I Semantics

I Turing-complete?

I Compilers

I ...

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Programming languages and tools

I Semantics

I Turing-complete?

I Compilers

I ...

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Quantum circuits vs Quantum programs

Example:Quantum walk on a circle with an absorbing boundary.

I Hd=span{|Li,|Ri}— thedirectionspace,|Liand|Riindicate directions left and right, respectively.

I Hp=span{|0i,|1i, ...,|n−1i}— thepositionspace with

orthonormal basis states, the vector|iidenotes position i for each 0≤i≤n−1.

I The state space of the walk — H =Hp⊗H_d.

I The initial state —|0i_p|Li_d.

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Quantum circuits vs Quantum programs

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Quantum circuits vs Quantum programs

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Quantum circuits vs Quantum programs

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Quantum circuits vs Quantum programs

I Each step of the walk:

1. Measure the system to see whether the current position is 1 (absorbing boundary). If “yes”, terminates; otherwise, continues:

M = {Myes= |1ih1| ⊗I_d, Mno =I−Myes};

2. A “coin-tossing”operator C= ^√¹

2

1 1

1 −1

is applied on the direction space;

3. A shift operator

S=

n−1

∑

i=0

|i 1ihi| ⊗ |LihL| +

n−1

∑

i=0

|i⊕1ihi| ⊗ |RihR|

is performed on the state space H.

I Question:How to specify it in the circuit language?

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Quantum circuits vs Quantum programs

M = {Myes= |1ih1| ⊗I_d, Mno=I−Myes};

2

1 1

1 −1

3. A shift operator

S=

n−1

∑

i=0

n−1

∑

i=0

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Quantum circuits vs Quantum programs

2

1 1

1 −1

3. A shift operator

S=

n−1

∑

i=0

n−1

∑

i=0

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Quantum circuits vs Quantum programs

2

1 1

1 −1

3. A shift operator

S=

n−1

∑

i=0

n−1

∑

i=0

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Quantum circuits vs Quantum programs

2

1 1

1 −1

3. A shift operator

S=

n−1

∑

i=0

n−1

∑

i=0

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Outline

Introduction

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Classical while-Language

S ::= _skip|u :=t|S₁; S2|if b then S1else S2fi

|while b do S od.

I Conditional statement can be generalised to case statement:

if G1→S₁

G2→S2

...

Gn →Sn

fi or more compactly:

if(_i·Gi→Si)_fi

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Quantum while-Language

I Alphabet: a countably infinite set Var of quantum variables q, q⁰, q0, q1, q2, ....

I Each quantum variable q∈Var has atypeHq(a Hilbert space), e.g.

Boolean= H₂(2−dimensional), integer= H_∞=

( _∞

n=−

∑

∞

α_n|ni:

∑

∞ n=−∞

|α_n|²<∞ )

.

I A quantumregisteris a finite sequence q=q1, ..., qnof distinct quantum variables. State Hilbert space:

H_q=

n O i=1

H_q_i.

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Quantum while-Language

( _∞

n=−

∑

∞

α_n|ni:

∑

∞ n=−∞

|α_n|²<∞ )

.

H_q=

n O i=1

H_q_i.

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Quantum while-Language

( _∞

n=−

∑

∞

α_n|ni:

∑

∞ n=−∞

|α_n|²<∞ )

.

H_q=

n O i=1

H_q_i.

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Syntax of Quantum Programs

S ::= skip|q := |0i |q :=U[q] |S1; S2

|_if(_m·M[q] =m→Sm)_fi

|while M[q] =1 do S od.

Exercise 1

Write quantum walk as a program in quantum while-language.

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Syntax of Quantum Programs

S ::= skip|q := |0i |q :=U[q] |S1; S2

|_if(_m·M[q] =m→Sm)_fi

|while M[q] =1 do S od.

Exercise 1

Write quantum walk as a program in quantum while-language.

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Outline

Introduction

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Notations

I partial density operator: positive operator ρ, tr(ρ) ≤_1.

I D(H)— the set of partial density operators inH.

I State Hilbert space of all quantum variables: H_all = ^O

q∈Var

Hq.

I ↓— empty program; i.e. termination.

I Configuration: pairhS, ρi, where:

1. S is a quantum program or the empty program↓;

2. ρ∈ D(H_all), denoting the (global) state of quantum variables.

I Transition:

hS, ρi → hS⁰, ρ⁰i

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Notations

I State Hilbert space of all quantum variables: H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

I State Hilbert space of all quantum variables:

H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

H_all = ^O

q∈Var

Hq.

I Transition:

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Notations

H_all = ^O

q∈Var

Hq.

I Transition:

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Operational Semantics

(SK)

hskip, ρi → h↓, ρi

(IN)

hq := |₀i, ρi → h↓, ρ^q₀i

where : ρ^q₀=

∑

i

|0iqhi|ρ|iiqh0| (UT)

hq :=U[q], ρi → h↓, UρU^†i

(_SC) hS1, ρi → hS₁⁰, ρ⁰i hS1; S2, ρi → hS₁⁰; S2, ρ⁰i

where : ↓; S2=S2.

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Operational Semantics

(IF)

hif(m·M[q] =m→Sm)fi, ρi → hSm, MmρM^†_mi

for each possible outcome m of measurement M= {Mm}.

(L0)

hwhile M[q] =1 do S od, ρi → h↓, M0ρM^†₀i

(L1)

hwhile M[q] =1 do S od, ρi → hS; while M[q] =1 do S od, M1ρM^†₁i

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Computation of Programs

1. A (finite or infinite)transition sequenceof program S with input ρ∈ D(H_all):

hS, ρi → hS1, ρ1i →...→ hSn, ρni → hSn+1, ρn+1i →...

such that ρn , 0 for all n (except the last n in the case of a finite sequence).

2. If this sequence cannot be extended, it is called acomputation.

I If it is finite and its last configuration ish↓, ρ⁰i, we say it terminates in ρ⁰.

I If it is infinite, we say it diverges.

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Computation of Programs

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Computation of Programs

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Computation of Programs

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Notation

I

hS, ρi →ⁿ hS⁰, ρ⁰i

if there are configurationshS₁, ρ₁i, ...,hS_n−1, ρ_n−1isuch that hS, ρi → hS₁, ρ₁i →...→ hS_n−1, ρ_n−1i → hS⁰, ρ⁰i,

I →^∗for the reflexive and transitive closures of→: hS, ρi →^∗ hS⁰, ρ⁰i

if and only ifhS, ρi →ⁿ hS⁰, ρ⁰ifor some n≥0.

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Notation

I

hS, ρi →ⁿ hS⁰, ρ⁰i

if there are configurationshS₁, ρ₁i, ...,hS_n−1, ρ_n−1isuch that hS, ρi → hS₁, ρ₁i →...→ hS_n−1, ρ_n−1i → hS⁰, ρ⁰i,

I →^∗for the reflexive and transitive closures of→: hS, ρi →^∗ hS⁰, ρ⁰i

if and only ifhS, ρi →ⁿ hS⁰, ρ⁰ifor some n≥0.

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Outline

Introduction

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Semantic Function

Semantic function of program S:

~S :D(H_all) → D(H_all)

~S(ρ) =

∑

^|^ρ⁰ ^:^h^{S, ρ}^{i →}^∗ ^h↓^{, ρ}⁰^i|

Exercise 2

Try to compute semantic function of your quantum walk program.

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Semantic Function

Semantic function of program S:

~S :D(H_all) → D(H_all)

~S(ρ) =

∑

^|^ρ⁰ ^:^h^{S, ρ}^{i →}^∗ ^h↓^{, ρ}⁰^i|

Exercise 2

Try to compute semantic function of your quantum walk program.

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Structural Representation

1. ~skip(ρ) =ρ.

2. ~q := |0i(ρ) =_∑_i|0iqhi|ρ|iiqh0|. 3. ~q :=U[q](ρ) =UρU^†.

4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

5. ~if(_m·M[q] =m→Sm)_fi(ρ) =_∑_m_~S_m(MmρM^†_m). 6. ~while M[q] =_{1 do S od}(ρ) = _???

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Structural Representation

1. ~skip(ρ) =ρ.

2. ~q := |0i(ρ) =_∑_i|0iqhi|ρ|iiqh0|.

3. ~q :=U[q](ρ) =UρU^†. 4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

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Structural Representation

1. ~skip(ρ) =ρ.

4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

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Structural Representation

1. ~skip(ρ) =ρ.

4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

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Structural Representation

1. ~skip(ρ) =ρ.

4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

5. ~if(_m·M[q] =m→Sm)_fi(ρ) =_∑_m_~S_m(MmρM^†_m).

6. ~while M[q] =_{1 do S od}(ρ) = _???

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Structural Representation

1. ~skip(ρ) =ρ.

4. ~S1; S2(ρ) =_~S₂(_~S₁(ρ)).

5. ~if(_m·M[q] =m→Sm)_fi(ρ) =_∑_m_~S_m(MmρM^†_m). 6. ~while M[_q] =_{1 do S od}(ρ) = _???

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Basic Lattice Theory

I Apartial order(L,v): L is a nonempty set,vis a binary relation on L satisfying:

1. Reflexivity: xvx for all x∈_L;

2. Antisymmetry: xv_{y and y}v_{x imply x}=y for all x, y∈_L; 3. Transitivity: xv_{y and y}v_{z imply x}vz for all x, y, z∈_L.

I x∈L is called theleast elementwhen xvy for all y∈L.

I x∈L is called an upper bound of a subset X⊆L if yvx for all x∈X.

I x is called the least upper bound of X, written x=^FX, if

I x is an upper bound of X;

I for any upper bound y of X, xv_y.

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Basic Lattice Theory

2. Antisymmetry: xv_{y and y}v_{x imply x}=y for all x, y∈_L; 3. Transitivity: xv_{y and y}v_{z imply x}vz for all x, y, z∈_L.

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Basic Lattice Theory

2. Antisymmetry: xv_{y and y}v_{x imply x}=y for all x, y∈_L;

3. Transitivity: xv_{y and y}v_{z imply x}vz for all x, y, z∈_L.

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Basic Lattice Theory

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Basic Lattice Theory

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Basic Lattice Theory

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Basic Lattice Theory

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Basic Lattice Theory

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Basic Lattice Theory

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Basic Lattice Theory

I Acomplete latticeis a partial order(L,v)such that^FX exists (and so does

X) for any X⊆L.

I Acomplete partial order(CPO) is a partial order(L,v):

1. it has the least element 0;

2. ^F^∞_n=0xnexists for any increasing sequence{xn}: x₀v...vxnvx_n+1v....

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Basic Lattice Theory

X) for any X⊆L.

I Acomplete partial order(CPO) is a partial order(L,v):

1. it has the least element 0;

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Basic Lattice Theory

X) for any X⊆L.

I Acomplete partial order(CPO) is a partial order(L,v): 1. it has the least element 0;

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Basic Lattice Theory

X) for any X⊆L.

I Acomplete partial order(CPO) is a partial order(L,v): 1. it has the least element 0;

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Basic Lattice Theory

I A function f from L into itself is

I monotoneif:

xvy⇒f(x) vf(y);

I continuousif:

f ^G

n

x_n

!

=^G

n

f(x_n) for any increasing sequence{xn}in L.

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Basic Lattice Theory

I monotoneif:

xvy⇒f(x) vf(y);

I continuousif:

f ^G

n

x_n

!

=^G

n

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Basic Lattice Theory

I monotoneif:

xvy⇒f(x) vf(y);

I continuousif:

f ^G

n

x_n

!

=^G

n

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Tarski Fixed Point Theorem

I If(L,v)is a complete lattice and function f : L→L monotone, then f has a fixed point.

I Let(L,v)be a CPO and function f : L→L continuous. Then f has the least fixed point

µf = G∞

n=0

f⁽ⁿ⁾(0)

where f⁽⁰⁾(0) =0 and f⁽ⁿ⁺¹⁾(0) =f(f⁽ⁿ⁾(0))for n≥0.

Exercise 3

Use Tarski fixed point theorem to proveSchröder-Bernstein theorem: if f : X→Y and g : Y→X are one-to-one mappings, then there is a bijection between X and Y.

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Tarski Fixed Point Theorem

I If(L,v)is a complete lattice and function f : L→L monotone, then f has a fixed point.

I Let(L,v)be a CPO and function f : L→L continuous. Then f has the least fixed point

µf = G∞

n=0

f⁽ⁿ⁾(0)

where f⁽⁰⁾(0) =0 and f⁽ⁿ⁺¹⁾(0) =f(f⁽ⁿ⁾(0))for n≥0.

Exercise 3

Use Tarski fixed point theorem to proveSchröder-Bernstein theorem:

if f : X→Y and g : Y→X are one-to-one mappings, then there is a bijection between X and Y.

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CPO of Partial Density Operators

I Löwner order: operators AvB⇔B−A is positive.

I (D(H),v)is a CPO with the zero operator 0Has its least element.

Exercise 4

Prove the above statement forfinite-dimensionalH.

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CPO of Partial Density Operators

I Löwner order: operators AvB⇔B−A is positive.

I (D(H),v)is a CPO with the zero operator 0Has its least element.

Exercise 4

Prove the above statement forfinite-dimensionalH.

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CPO of Super-operators

I Each super-operator inHis a continuous function from (D(H),v)into itself.

I QO(H)— the set of superoperators in Hilbert spaceH.

I Löwner order between operators can be lifted to a partial order between super-operators:

E v F ⇔ E (ρ) v F (ρ)for all ρ∈ D(H)

I (QO(H),v)is a CPO.

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CPO of Super-operators

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CPO of Super-operators

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CPO of Super-operators

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Syntactic Approximation

I abortdenotes a program such that

~abort(ρ) =0H_allfor all ρ∈ D(H).

I Write:

while≡while M[q] =1 do S od.

I For integer k≥0, the kth syntactic approximation while^(k)of while:











while⁽⁰⁾ ≡ abort,

while^(k+1) ≡ if M[q] =0→skip

1→S; while^(k) fi

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Syntactic Approximation

I Write:











while^(k+1) ≡ if M[q] =0→skip

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Syntactic Approximation

I Write:











while^(k+1) ≡ if M[q] =0→skip

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Semantic Function of Loops

while

= G∞

k=0

while^(k) ,

where^Fstands for the least upper bound in CPO(QO (H_all),v).

Fixed Point Characterisation

For any ρ∈ D(H_all):

~while(ρ) =M0ρM^†₀+_~while_~SM1ρM₁^†

.

Exercise 5

Prove the above equality.

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Semantic Function of Loops

while

= G∞

k=0

while^(k) ,

Fixed Point Characterisation

.

Exercise 5

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Semantic Function of Loops

while

= G∞

k=0

while^(k) ,

Fixed Point Characterisation

.

Exercise 5

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Termination and Divergence Probabilities

I For any quantum program S and ρ∈ D(H_all): tr(_~S(ρ)) ≤tr(ρ).

I tr(_~S(ρ))is the probability that program S with input ρ terminates.

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Termination and Divergence Probabilities

I For any quantum program S and ρ∈ D(H_all): tr(_~S(ρ)) ≤tr(ρ).

I tr(_~S(ρ))is the probability that program S with input ρ terminates.

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Outline

Introduction

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Quantum predicates = Quantum effects

I Aquantum effectis a (Hermitian) operator 0vMvI; that is, 0≤tr(_Mρ) ≤1 for all density operators.

I tr(Mρ)may be interpreted as the expected degree to which quantum state ρ satisfies quantum predicate M.

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Quantum predicates = Quantum effects

I Aquantum effectis a (Hermitian) operator 0vMvI; that is, 0≤tr(_Mρ) ≤1 for all density operators.

I tr(Mρ)may be interpreted as the expected degree to which quantum state ρ satisfies quantum predicate M.

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Correctness Formulas

I A correctness formula (Hoare triple) is a statement of the form:

{P}S{Q} where:

I S is a quantum program;

I P, Q are quantum predicates inH_all.

I P is called the precondition, Q the postcondition.

Partial Correctness, Total Correctness

I Partial correctness: If an input to program S satisfies precondition P, then either S does not terminate, or it terminates in a state satisfying postcondition Q.

I Total correctness: If an input to program S satisfies precondition P, then S must terminate and it terminates in a state satisfying postcondition Q.

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Correctness Formulas

{P}S{Q} where:

Partial Correctness, Total Correctness

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Correctness Formulas

{P}S{Q} where:

Partial Correctness, Total Correctness

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Correctness Formulas

{P}S{Q} where:

Partial Correctness, Total Correctness

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Correctness Formulas

{P}S{Q} where:

Partial Correctness, Total Correctness

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Correctness Formulas

{P}S{Q} where:

Partial Correctness, Total Correctness

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Partial Correctness, Total Correctness (Continued)

I The correctness formula{P}S{Q}is true in the sense of total correctness, written

|=_tot{P}S{Q}, if:

tr(Pρ) ≤tr(_Q~S(ρ))

for all ρ∈ D(H_all), where ~S is the semantic function of S.

I The correctness formula{P}S{Q}is true in the sense of partial correctness, written

|=_par{P}S{Q}, if:

tr(Pρ) ≤tr(_Q~S(ρ)) + [tr(ρ) −tr(_~S(ρ))] for all ρ∈ D(H_all).

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Partial Correctness, Total Correctness (Continued)

I The correctness formula{P}S{Q}is true in the sense of total correctness, written

|=_tot{P}S{Q}, if:

tr(Pρ) ≤tr(_Q~S(ρ))

for all ρ∈ D(H_all), where ~S is the semantic function of S.

I The correctness formula{P}S{Q}is true in the sense of partial correctness, written

|=_par {P}S{Q}, if:

tr(Pρ) ≤tr(_Q~S(ρ)) + [tr(ρ) −tr(_~S(ρ))]

for all ρ∈ D(H_all).

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Proof System for Partial Correctness

(Ax−Sk) {P}Skip{P} (Ax−In)If type(q) =Boolean, then

{|0i_qh0|P|0i_qh0| + |1i_qh0|P|0i_qh1|}q := |0i{P}

If type(q) =integer, then

( _∞

n=−∞

∑

|ni_qh0|P|0i_qhn| )

q := |0i{P}

(Ax−UT) {U^†PU}q :=Uq{P}

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Proof System for Partial Correctness (Continued)

(R−SC) {P}S₁{Q} {Q}S₂{R} {P}S₁; S2{R}

(R−IF) {Pm}Sm{Q}for all m

∑mM^†_mPmMm if(_m·M[q] =m→Sm)_fi{Q}

(R−LP) {Q}SM^†₀PM0+M^†₁QM1

{M^†₀PM₀+M^†₁QM₁}_{while M}[q] =1 do S od{P}

(R−Or) ^PvP⁰ {P⁰}S{Q⁰} Q⁰vQ {P}S{Q}

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Soundness Theorem

For any quantum while-program S and quantum predicates P, Q∈ P (H_all)_:

`_qPD{P}S{Q}implies |=_par {P}S{Q}.

Exercise 6

Prove soundness theorem.

(Relative) Completeness Theorem

For any quantum while-program S and quantum predicates P, Q∈ P (H_all):

|=_par{P}S{Q}implies `_qPD{P}S{Q}.

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Soundness Theorem

For any quantum while-program S and quantum predicates P, Q∈ P (H_all)_:

`_qPD{P}S{Q}implies |=_par {P}S{Q}.

Exercise 6

Prove soundness theorem.

(Relative) Completeness Theorem

For any quantum while-program S and quantum predicates P, Q∈ P (H_all):

|=_par{P}S{Q}implies `_qPD{P}S{Q}.