Clifford group from scratch Maris Ozols University of Waterloo - - PowerPoint PPT Presentation

Introduction Definition Single qubit case Order Generators Applications References

Clifford group from scratch

Maris Ozols

University of Waterloo

July 28, 2008

Introduction Definition Single qubit case Order Generators Applications References

Outline

1 Introduction 2 Definition of the Clifford group Cn on n qubits 3 Clifford group C1 of a single qubit 4 Number of elements in Cn 5 Generators of Cn 6 Applications

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is,

• nly Maris doesn’t know. . .

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is,

• nly Maris doesn’t know. . .

I’m obsessed with symmetric structures in the Hilbert space

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is,

• nly Maris doesn’t know. . .

I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is,

• nly Maris doesn’t know. . .

I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications I know the results, but I haven’t seen the proofs

Introduction Definition Single qubit case Order Generators Applications References

Motivation

Everybody knows what the Clifford group is,

• nly Maris doesn’t know. . .

I’m obsessed with symmetric structures in the Hilbert space Clifford group has lots of applications I know the results, but I haven’t seen the proofs Some folklore results with no proofs available

Introduction Definition Single qubit case Order Generators Applications References

Pauli matrices

Single qubit

The set of Pauli matrices is P = {I, X, Y, Z}, where I = 1 0

0 1

• ,

X = 0 1

1 0

• ,

Y = 0 −i

i 0

• ,

Z = 1 0

0 −1

• .

Introduction Definition Single qubit case Order Generators Applications References

Pauli matrices

Single qubit

The set of Pauli matrices is P = {I, X, Y, Z}, where I = 1 0

0 1

• ,

X = 0 1

1 0

• ,

Y = 0 −i

i 0

• ,

Z = 1 0

0 −1

• .

For n qubits

Pn = {σ1 ⊗ σ2 ⊗ · · · ⊗ σn | σi ∈ P} .

Introduction Definition Single qubit case Order Generators Applications References

Pauli matrices

Single qubit

The set of Pauli matrices is P = {I, X, Y, Z}, where I = 1 0

0 1

• ,

X = 0 1

1 0

• ,

Y = 0 −i

i 0

• ,

Z = 1 0

0 −1

• .

For n qubits

Pn = {σ1 ⊗ σ2 ⊗ · · · ⊗ σn | σi ∈ P} .

Vector space structure

The group Pn/U(1) is isomorphic to a vector space over F2 with dimension 2n via identification Z Y | | I X multiply ⇐ ⇒ (0, 1) (1, 1) | | (0, 0) (1, 0) add

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in P ∗

n have eigenvalues ±1 with equal multiplicity.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X,

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X,

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X,

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I,

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I, X → iX.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take Paulis to Paulis via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I, X → iX.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take ±P ∗

n to ±P ∗ n via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I, X → iX.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition (sloppy)

Unitaries that take ±P ∗

n to ±P ∗ n via conjugation.

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I, X → iX.

Global phase

U and eiϕU act identically, i.e., UMU† = (eiϕU)M(eiϕU)†.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Definition

The Clifford group Cn on n qubits is Cn =

• U ∈ U(2n) | σ ∈ ±P ∗

n ⇒ UσU† ∈ ±P ∗ n

• /U(1).

Eigenvalues

The eigenvalues of X, Y , Z are ±1. Let P ∗

n = Pn \ {I⊗n}.

All matrices in ± P ∗

n have eigenvalues ±1 with equal multiplicity.

You can

X → −X, e.g., ZXZ = −X, X ⊗ I → X ⊗ X, e.g., CNOT(X ⊗ I)CNOT † = X ⊗ X.

You cannot

X → I, X → iX.

Global phase

U and eiϕU act identically, i.e., UMU† = (eiϕU)M(eiϕU)†.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†),

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†), U(−X)U† = −UXU † and similarly for Z.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†), U(−X)U† = −UXU † and similarly for Z. Thus it is enough to specify where X and Z go.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†), U(−X)U† = −UXU † and similarly for Z. Thus it is enough to specify where X and Z go. However, since X and Z anti-commute, so must UXU† and UZU†.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†), U(−X)U† = −UXU † and similarly for Z. Thus it is enough to specify where X and Z go. However, since X and Z anti-commute, so must UXU† and UZU†.

All possibilities

X can go to any element of ±P ∗

1 ,

Z can go to any element of ±P ∗

1 \

• ±UXU†

.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Single qubit

±P ∗

1 = {±X, ±Y, ±Z}.

Restrictions

Conjugation must preserve the structure of Pauli matrices. Y = iXZ, thus UY U† = i(UXU†)(UZU†), U(−X)U† = −UXU † and similarly for Z. Thus it is enough to specify where X and Z go. However, since X and Z anti-commute, so must UXU† and UZU†.

All possibilities

X can go to any element of ±P ∗

1 ,

Z can go to any element of ±P ∗

1 \

• ±UXU†

.

Group order

|C1| = 6 · 4 = 24.

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Clifford group rotations

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Clifford group rotations

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Clifford group rotations

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Clifford group rotations

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Cuboctahedron

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Cuboctahedron

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Cuboctahedron Poll

Guess what’s the value of |C2|?

Introduction Definition Single qubit case Order Generators Applications References

Clifford group C1

Cuboctahedron Poll

Guess what’s the value of |C2|? Answer: |C2| = 11520.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

It is enough to specify where Xi and Zi go for all i ∈ {1, . . . , n}.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

It is enough to specify where Xi and Zi go for all i ∈ {1, . . . , n}. All X’s and Z’s commute, except Xi and Zi that anti-commute: X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

It is enough to specify where Xi and Zi go for all i ∈ {1, . . . , n}. All X’s and Z’s commute, except Xi and Zi that anti-commute: X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Claim

Each matrix in ±P ∗

n commutes (anti-commutes) with exactly half

• f Pauli matrices Pn.

Proof.

Let σ ∈ ±P ∗

n and k be a position where σ does not contain I. All

Paulis that anti-commute with σ can be constructed as follows:

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

It is enough to specify where Xi and Zi go for all i ∈ {1, . . . , n}. All X’s and Z’s commute, except Xi and Zi that anti-commute: X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Claim

Each matrix in ±P ∗

n commutes (anti-commutes) with exactly half

• f Pauli matrices Pn.

Proof.

Let σ ∈ ±P ∗

n and k be a position where σ does not contain I. All

Paulis that anti-commute with σ can be constructed as follows: put any of I, X, Y , Z at each position other than k,

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

It is enough to specify where Xi and Zi go for all i ∈ {1, . . . , n}. All X’s and Z’s commute, except Xi and Zi that anti-commute: X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Claim

Each matrix in ±P ∗

n commutes (anti-commutes) with exactly half

• f Pauli matrices Pn.

Proof.

Let σ ∈ ±P ∗

n and k be a position where σ does not contain I. All

Paulis that anti-commute with σ can be constructed as follows: put any of I, X, Y , Z at each position other than k, fill the kth position in any of two possible ways so that the

• btained matrix anti-commutes with σ.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Counting

Where can U ∈ Cn send Xn and Zn?

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Counting

Where can U ∈ Cn send Xn and Zn? Xn can go to any element of ±P ∗

n, i.e., 2(4n − 1) choices,

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Counting

Where can U ∈ Cn send Xn and Zn? Xn can go to any element of ±P ∗

n, i.e., 2(4n − 1) choices,

Zn can go to any element of ±P ∗

n that anti-commutes with

UXnU†, i.e., 2|Pn|

2

= 4n choices.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Counting

Where can U ∈ Cn send Xn and Zn? Xn can go to any element of ±P ∗

n, i.e., 2(4n − 1) choices,

Zn can go to any element of ±P ∗

n that anti-commutes with

UXnU†, i.e., 2|Pn|

2

= 4n choices. Similarly for the next pair (Xn−1, Zn−1), just replace n by n − 1.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Restrictions

X1 X2 . . . Xn−1 Xn | | | | Z1 Z2 . . . Zn−1 Zn

Counting

Where can U ∈ Cn send Xn and Zn? Xn can go to any element of ±P ∗

n, i.e., 2(4n − 1) choices,

Zn can go to any element of ±P ∗

n that anti-commutes with

UXnU†, i.e., 2|Pn|

2

= 4n choices. Similarly for the next pair (Xn−1, Zn−1), just replace n by n − 1.

Result

|Cn| =

n

• j=1

2(4j − 1) · 4j = 2n2+2n

n

• j=1

(4j − 1).

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

How does it grow?

n |Cn| 1 24 2 11520 3 92897280 4 12128668876800 5 25410822678459187200

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

How does it grow?

n |Cn| 1 24 2 11520 3 92897280 4 12128668876800 5 25410822678459187200 This is 1

8 times “Sloane’s A003956”.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

How does it grow?

n |Cn| 1 24 2 11520 3 92897280 4 12128668876800 5 25410822678459187200 This is 1

8 times “Sloane’s A003956”.

Upper bound

|Cn| ≤ 2n2+2n

n

• j=1

4j = 22n2+3n.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Their definition

Calderbank R.A., Rains E.M., Shor P.W., Sloane N.J.A., Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006v5.

Introduction Definition Single qubit case Order Generators Applications References

Order of Cn

Their definition

Calderbank R.A., Rains E.M., Shor P.W., Sloane N.J.A., Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006v5.

Explanation of factor 8

They assume that H, P ∈ Cn, i.e., they define Cn as the group generated by H, P, and CNOT. Thus they get 8 times more, since ηI ∈ Cn, where η = 1+i

√ 2 is the 8th root of unity.

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Theorem

The Clifford group Cn is generated by H, P, and CNOT: H =

1 √ 2

1 1

1 −1

• ,

P = 1 0

0 i

• ,

CNOT = 1 0 0 0

0 1 0 0 0 0 0 1 0 0 1 0

• .

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Theorem

The Clifford group Cn is generated by H, P, and CNOT: H =

1 √ 2

1 1

1 −1

• ,

P = 1 0

0 i

• ,

CNOT = 1 0 0 0

0 1 0 0 0 0 0 1 0 0 1 0

• .

More precisely, Cn = Hi, Pi, CNOTij /U(1).

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Theorem

The Clifford group Cn is generated by H, P, and CNOT: H =

1 √ 2

1 1

1 −1

• ,

P = 1 0

0 i

• ,

CNOT = 1 0 0 0

0 1 0 0 0 0 0 1 0 0 1 0

• .

More precisely, Cn = Hi, Pi, CNOTij /U(1).

Proof.

It is easy to verify that C1 = H, P /U(1). Use induction on n.

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Proof (continued).

Let U ∈ Cn+1. Since X1 and Z1 anti-commute, so do UX1U† and UZ1U†. We can permute qubits and apply elements of C1 so that UX1U† = X ⊗ M′, UZ1U† = Z ⊗ N′. for some M′, N′ ∈ ±Pn.

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Proof (continued).

Let U ∈ Cn+1. Since X1 and Z1 anti-commute, so do UX1U† and UZ1U†. We can permute qubits and apply elements of C1 so that UX1U† = X ⊗ M′, UZ1U† = Z ⊗ N′. for some M′, N′ ∈ ±Pn. Let U(|0 ⊗ |ψ) =

1 √ 2(|0 ⊗ |ψ0 + |1 ⊗ |ψ1).

Define U′ by U′ |ψ = |ψ0. One can show that U′ ∈ Cn.

Introduction Definition Single qubit case Order Generators Applications References

Generators of Cn

Proof (continued).

Let U ∈ Cn+1. Since X1 and Z1 anti-commute, so do UX1U† and UZ1U†. We can permute qubits and apply elements of C1 so that UX1U† = X ⊗ M′, UZ1U† = Z ⊗ N′. for some M′, N′ ∈ ±Pn. Let U(|0 ⊗ |ψ) =

1 √ 2(|0 ⊗ |ψ0 + |1 ⊗ |ψ1).

Define U′ by U′ |ψ = |ψ0. One can show that U′ ∈ Cn. Then we can implement U as follows:

• H
• /

U′ / N′ / M′ /

Introduction Definition Single qubit case Order Generators Applications References

Stabilizer formalism

Introduction Definition Single qubit case Order Generators Applications References

Stabilizer formalism

Who doesn’t know that the stabilizer formalism is?

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Schr¨

• dinger vs. Heisenberg

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Schr¨

• dinger vs. Heisenberg

Schr¨

• dinger picture: quantum states evolve in time,

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Schr¨

• dinger vs. Heisenberg

Schr¨

• dinger picture: quantum states evolve in time,

Heisenberg picture: operators evolve in time.

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Schr¨

• dinger vs. Heisenberg

Schr¨

• dinger picture: quantum states evolve in time,

Heisenberg picture: operators evolve in time.

Theorem (Gottesman-Knill)

Any quantum computation involving only: measurements in standard basis, Clifford group gates (conditioned on classical bits, e.g., measurement outcomes) can be perfectly simulated in polynomial time on a probabilistic classical computer.

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Schr¨

• dinger vs. Heisenberg

Schr¨

• dinger picture: quantum states evolve in time,

Heisenberg picture: operators evolve in time.

Theorem (Gottesman-Knill)

Any quantum computation involving only: measurements in standard basis, Clifford group gates (conditioned on classical bits, e.g., measurement outcomes) can be perfectly simulated in polynomial time on a probabilistic classical computer.

CHP (CNOT-Hadamard-Phase)

Program in C written by Aaronson and Gottesman to simulate such circuits. Can easily handle up to 3000 qubits!

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . .

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . .

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . . natural module . . .

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . . natural module . . . inertia group . . .

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . . natural module . . . inertia group . . . is ramified . . .

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . . natural module . . . inertia group . . . is ramified . . . which is a contradiction.

Introduction Definition Single qubit case Order Generators Applications References

Universal set of quantum gates

Mathematicians have shown that. . .

Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2.

In other words

Let m ≥ 1. Then Cm together with any other gate not in Cm form a universal set of quantum gates.

Proof.

. . . RCM-lattices . . . natural module . . . inertia group . . . is ramified . . . which is a contradiction.

Question

Is there an elementary proof for this?

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Universal set of quantum gates

Another zero-knowledge proof

Introduction Definition Single qubit case Order Generators Applications References

References

Introduction Definition Single qubit case Order Generators Applications References

Clifford group

Calderbank R.A., Rains E.M., Shor P.W., Sloane N.J.A., Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006v5. Nebe G., Rains E.M., Sloane N.J.A., The Invariants of the Clifford Groups, arXiv:math/0001038v2. Planat M., Jorrand P., Group theory for quantum gates and quantum coherence, arXiv:0803.1911v2 [quant-ph].

Introduction Definition Single qubit case Order Generators Applications References

H, P, and CNOT generate Cn

Gottesman D., Stabilizer Codes and Quantum Error Correction, PhD thesis, arXiv:quant-ph/9705052v1. Gottesman D., A Theory of Fault-Tolerant Quantum Computation, arXiv:quant-ph/9702029v2. Nielsen M.A., Chuang I.L., Quantum Computation and Quantum Information, Cambridge University Press, 2000.

Introduction Definition Single qubit case Order Generators Applications References

Gottesman-Knill theorem

Gottesman D., The Heisenberg Representation of Quantum Computers, arXiv:quant-ph/9807006v1. Aaronson S., Gottesman D., Improved Simulation of Stabilizer Circuits, arXiv:quant-ph/0406196v5.

Introduction Definition Single qubit case Order Generators Applications References

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