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Synthesis of Logical Clifford Operators via Symplectic Geometry Narayanan Rengaswamy Information Initiative at Duke (iiD), Duke University Joint Work: Swanand Kadhe, Robert Calderbank, and Henry Pfister 2018 IEEE Intl. Symp. on Information

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SLIDE 1

Synthesis of Logical Clifford Operators via Symplectic Geometry

Narayanan Rengaswamy Information Initiative at Duke (iiD), Duke University Joint Work: Swanand Kadhe, Robert Calderbank, and Henry Pfister 2018 IEEE Intl. Symp. on Information Theory Vail, Colorado, USA arXiv:1803.06987 June 19, 2018

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 1 / 21

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SLIDE 2

Overview

1

Motivation and our Contribution

2

Essential Algebraic Setup

3

Synthesis of Logical Clifford Operators for Stabilizer Codes

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 2 / 21

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SLIDE 3

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 1 m1 ⊕ m2 ⊕ m3 m3 ⊕ 1 m2 m1 ⊕ m2

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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SLIDE 4

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 X 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 1 m1 ⊕ m2 ⊕ m3 m3 ⊕ 1 m2 m1 ⊕ m2

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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SLIDE 5

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 X 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 1 m1 ⊕ m2 ⊕ m3 m3 ⊕ 1 m2 m1 ⊕ m2

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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SLIDE 6

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 X 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 1 m1 ⊕ m3 ⊕ 1 m3 ⊕ 1 m2 m1 ⊕ m2 KEY: Physical circuit has to realize logical operation & preserve the code.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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SLIDE 7

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 X 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 X 1 m1 ⊕ m3 ⊕ 1 m3 ⊕ 1 m2 m1 ⊕ m2 KEY: Physical circuit has to realize logical operation & preserve the code.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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SLIDE 8

Encoded Computation: An Example

Uncoded: 3 bits m1 m2 m3 1 X 1 m3 ⊕ 1 m2 m1 ⊕ m2 Coded: [4, 3, 2] SPC m1 m2 m3 1 X 1 m1 ⊕ m3 ⊕ 1 m3 ⊕ 1 m2 m1 ⊕ m2 KEY: Physical circuit has to realize logical operation & preserve the code. In the quantum analogue of this, we have arbitrary unitary operators!

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21

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Problem and Motivation

Quantum systems are noisy - quantum error-correcting codes (QECCs) and quantum control are essential for reliable computation. QECCs encode logical information into physical states. Lots of interesting work on QECCs, their properties and efficient decoders. QECCs alone aren’t enough; need to perform computation on the protected information stored in physical qubits. Synthesis of physical operators that realize such encoded computation seems to exist essentially for particular QECC examples. We propose a systematic framework for synthesizing a large class of such operators, called the Clifford group, for stabilizer QECCs.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 4 / 21

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SLIDE 10

Problem and Motivation

Quantum systems are noisy - quantum error-correcting codes (QECCs) and quantum control are essential for reliable computation. QECCs encode logical information into physical states. Lots of interesting work on QECCs, their properties and efficient decoders. QECCs alone aren’t enough; need to perform computation on the protected information stored in physical qubits. Synthesis of physical operators that realize such encoded computation seems to exist essentially for particular QECC examples. We propose a systematic framework for synthesizing a large class of such operators, called the Clifford group, for stabilizer QECCs.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 4 / 21

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SLIDE 11

Problem and Motivation

Quantum systems are noisy - quantum error-correcting codes (QECCs) and quantum control are essential for reliable computation. QECCs encode logical information into physical states. Lots of interesting work on QECCs, their properties and efficient decoders. QECCs alone aren’t enough; need to perform computation on the protected information stored in physical qubits. Synthesis of physical operators that realize such encoded computation seems to exist essentially for particular QECC examples. We propose a systematic framework for synthesizing a large class of such operators, called the Clifford group, for stabilizer QECCs.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 4 / 21

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SLIDE 12

Problem: Operations on Encoded Qubits

|xL |˜ xL |ψx |ψ˜

x

arbitrary logical operation QECC encode relevant physical operation QECC decode initial logical state desired final state unreliable circuit

QECC: Quantum Error-Correcting Codes

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 5 / 21

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SLIDE 13

Problem: Operations on Encoded Qubits

|xL |˜ xL |ψx |ψ˜

x

arbitrary logical operation QECC encode relevant physical operation QECC decode Need to translate for given QECC initial logical state desired final state unreliable circuit We do this for logical Clifford operations on stabilizer QECCs

QECC: Quantum Error-Correcting Codes

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 5 / 21

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Algorithms on GitHub

Our algorithms are available open-source at: https://github.com/nrenga/symplectic-arxiv18a Our Algorithm Stabilizer S (defines the code) Logical Paulis ¯ Xi, ¯ Zi [Got97; Wil09] Logical Clifford Operator gL All physical circuits ¯ g that realize gL & fix S

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 6 / 21

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Overview

1

Motivation and our Contribution

2

Essential Algebraic Setup

3

Synthesis of Logical Clifford Operators for Stabilizer Codes

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 7 / 21

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SLIDE 16

Pure States

Qubit: Mathematically, it is a 2-dimensional Hilbert space over C. Pure state: |ψ = α |0 + β |1 , with α, β ∈ C and |α|2 + |β|2 = 1.

Example (m = 2 qubits) : |0 ⊗ |1 = 1

  • ⊗

1

  • =

    1     = |01 .

m Qubits: If qubit i is in the state |vi ∈ {|0 , |1} then the Kronecker product |v1 ⊗ · · · ⊗ |vm |v describes the state of the system. Note that CN = C2m = C2 ⊗ · · · ⊗ C2 (m times). N = 2m. Pure state (m qubits): |φ =

v∈Fm

2 αv |v, αv ∈ C,

v∈Fm

2 |αv|2 = 1. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 8 / 21

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SLIDE 17

Pure States

Qubit: Mathematically, it is a 2-dimensional Hilbert space over C. Pure state: |ψ = α |0 + β |1 , with α, β ∈ C and |α|2 + |β|2 = 1.

Example (m = 2 qubits) : |0 ⊗ |1 = 1

  • ⊗

1

  • =

    1     = |01 .

m Qubits: If qubit i is in the state |vi ∈ {|0 , |1} then the Kronecker product |v1 ⊗ · · · ⊗ |vm |v describes the state of the system. Note that CN = C2m = C2 ⊗ · · · ⊗ C2 (m times). N = 2m. Pure state (m qubits): |φ =

v∈Fm

2 αv |v, αv ∈ C,

v∈Fm

2 |αv|2 = 1. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 8 / 21

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Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit: HW2 ικ{I2, X, Z, Y }, ι √ −1, κ ∈ {0, 1, 2, 3}. Bit-Flip: X 1 1

  • ⇒ X |0 = |1 , X |1 = |0 .

Phase-Flip: Z 1 −1

  • ⇒ Z |0 = |0 , Z |1 = − |1 .

Bit-Phase Flip: Y −ι ι

  • = ιXZ. XZ = −ZX.

Fact: {I2, X, Z, Y } forms an orthonormal basis for operators in C2×2. For m Qubits: HWN Kronecker products of m HW2 matrices (N = 2m).

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21

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Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit: HW2 ικ{I2, X, Z, Y }, ι √ −1, κ ∈ {0, 1, 2, 3}. Bit-Flip: X 1 1

  • ⇒ X |0 = |1 , X |1 = |0 .

Phase-Flip: Z 1 −1

  • ⇒ Z |0 = |0 , Z |1 = − |1 .

Bit-Phase Flip: Y −ι ι

  • = ιXZ. XZ = −ZX.

Fact: {I2, X, Z, Y } forms an orthonormal basis for operators in C2×2. For m Qubits: HWN Kronecker products of m HW2 matrices (N = 2m).

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21

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SLIDE 20

Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit: HW2 ικ{I2, X, Z, Y }, ι √ −1, κ ∈ {0, 1, 2, 3}. Bit-Flip: X 1 1

  • ⇒ X |0 = |1 , X |1 = |0 .

Phase-Flip: Z 1 −1

  • ⇒ Z |0 = |0 , Z |1 = − |1 .

Bit-Phase Flip: Y −ι ι

  • = ιXZ. XZ = −ZX.

Fact: {I2, X, Z, Y } forms an orthonormal basis for operators in C2×2. For m Qubits: HWN Kronecker products of m HW2 matrices (N = 2m).

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21

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Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit: HW2 ικ{I2, X, Z, Y }, ι √ −1, κ ∈ {0, 1, 2, 3}. Bit-Flip: X 1 1

  • ⇒ X |0 = |1 , X |1 = |0 .

Phase-Flip: Z 1 −1

  • ⇒ Z |0 = |0 , Z |1 = − |1 .

Bit-Phase Flip: Y −ι ι

  • = ιXZ. XZ = −ZX.

Fact: {I2, X, Z, Y } forms an orthonormal basis for operators in C2×2. For m Qubits: HWN Kronecker products of m HW2 matrices (N = 2m).

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21

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Hierarchy of the Unitary Group (m Qubits)

S HWN CliffN UN k-dimensional commutative subgroup of HWN – their common eigenspace defines a stabilizer code V (S) {|ψ : g |ψ = |ψ ∀ g ∈ S} HW2 ⊗ HW2 ⊗ · · · ⊗ HW2 (m times) ≈ F2m

2

under the symplectic inner product Maps every HWN element to another under conjugation ≈ Sp(2m, F2): 2m × 2m binary symplectic matrices The group of all 2m × 2m unitary matrices (N = 2m) CliffN: complexity reduced from 2m × 2m (complex) to 2m × 2m (binary)!

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 10 / 21

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SLIDE 23

Hierarchy of the Unitary Group (m Qubits)

S HWN CliffN UN k-dimensional commutative subgroup of HWN – their common eigenspace defines a stabilizer code V (S) {|ψ : g |ψ = |ψ ∀ g ∈ S} HW2 ⊗ HW2 ⊗ · · · ⊗ HW2 (m times) ≈ F2m

2

under the symplectic inner product Maps every HWN element to another under conjugation ≈ Sp(2m, F2): 2m × 2m binary symplectic matrices The group of all 2m × 2m unitary matrices (N = 2m) CliffN: complexity reduced from 2m × 2m (complex) to 2m × 2m (binary)!

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 10 / 21

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SLIDE 24

Hierarchy of the Unitary Group (m Qubits)

S HWN CliffN UN k-dimensional commutative subgroup of HWN – their common eigenspace defines a stabilizer code V (S) {|ψ : g |ψ = |ψ ∀ g ∈ S} HW2 ⊗ HW2 ⊗ · · · ⊗ HW2 (m times) ≈ F2m

2

under the symplectic inner product Maps every HWN element to another under conjugation ≈ Sp(2m, F2): 2m × 2m binary symplectic matrices The group of all 2m × 2m unitary matrices (N = 2m) CliffN: complexity reduced from 2m × 2m (complex) to 2m × 2m (binary)!

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 10 / 21

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SLIDE 25

Hierarchy of the Unitary Group (m Qubits)

S HWN CliffN UN k-dimensional commutative subgroup of HWN – their common eigenspace defines a stabilizer code V (S) {|ψ : g |ψ = |ψ ∀ g ∈ S} HW2 ⊗ HW2 ⊗ · · · ⊗ HW2 (m times) ≈ F2m

2

under the symplectic inner product Maps every HWN element to another under conjugation ≈ Sp(2m, F2): 2m × 2m binary symplectic matrices The group of all 2m × 2m unitary matrices (N = 2m) CliffN: complexity reduced from 2m × 2m (complex) to 2m × 2m (binary)!

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 10 / 21

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SLIDE 26

Heisenberg-Weyl and Clifford Groups

HWN Elements ≈ F2m

2 : γ(D(a, b)) [a, b]

Given binary m-tuples a = (a1, . . . , am), b = (b1, . . . , bm) define the matrix D(a, b) X a1Z b1 ⊗ · · · ⊗ X amZ bm ∈ UN ; N = 2m. ◮ D(a, b), D(a′, b′) ∈ HWN commute iff aTb′ + bTa′ = 0.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 11 / 21

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SLIDE 27

Heisenberg-Weyl and Clifford Groups

HWN Elements ≈ F2m

2 : γ(D(a, b)) [a, b]

Given binary m-tuples a = (a1, . . . , am), b = (b1, . . . , bm) define the matrix D(a, b) X a1Z b1 ⊗ · · · ⊗ X amZ bm ∈ UN ; N = 2m. ◮ D(a, b), D(a′, b′) ∈ HWN commute iff aTb′ + bTa′ = 0. CliffN Elements ≈ Sp(2m, F2): φ(g) Fg Define E(a, b) ιabT D(a, b). If g ∈ CliffN then gE(a, b)g† = ±E ([a, b]Fg) , where Fg is symplectic, i.e., satisfies FgΩF T

g = Ω =

Im Im

  • .

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 11 / 21

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SLIDE 28

Overview

1

Motivation and our Contribution

2

Essential Algebraic Setup

3

Synthesis of Logical Clifford Operators for Stabilizer Codes

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 12 / 21

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SLIDE 29

Synthesizing Logical CZ12 for the [ [6, 4, 2] ] Code

Definition (partial) : CZ12 ¯ X1CZ

† 12 ¯

X1 ¯ Z2, CZ12 ¯ X2CZ

† 12 ¯

Z1 ¯ X2. The process:

1 Generator matrices for the [6, 5, 2] SPC code yield logical Paulis:

¯ X1 = X1X2 = E(110000, 000000) ¯ Z1 = Z2Z6 = E(000000, 010001) ¯ X2 = X1X3 = E(101000, 000000) ¯ Z2 = Z3Z6 = E(000000, 001001)

2 Cliff26 ∼

= Sp(12, F2): CZ12E(a, b)CZ

† 12 = ±E

  • [a, b]FCZ12

. Find FCZ12. ¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001].

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 13 / 21

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SLIDE 30

Synthesizing Logical CZ12 for the [ [6, 4, 2] ] Code

Definition (partial) : CZ12 ¯ X1CZ

† 12 ¯

X1 ¯ Z2, CZ12 ¯ X2CZ

† 12 ¯

Z1 ¯ X2. The process:

1 Generator matrices for the [6, 5, 2] SPC code yield logical Paulis:

¯ X1 = X1X2 = E(110000, 000000) ¯ Z1 = Z2Z6 = E(000000, 010001) ¯ X2 = X1X3 = E(101000, 000000) ¯ Z2 = Z3Z6 = E(000000, 001001)

2 Cliff26 ∼

= Sp(12, F2): CZ12E(a, b)CZ

† 12 = ±E

  • [a, b]FCZ12

. Find FCZ12. ¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001].

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 13 / 21

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SLIDE 31

Synthesizing Logical CZ12 for the [ [6, 4, 2] ] Code

Definition (partial) : CZ12 ¯ X1CZ

† 12 ¯

X1 ¯ Z2, CZ12 ¯ X2CZ

† 12 ¯

Z1 ¯ X2. The process:

1 Generator matrices for the [6, 5, 2] SPC code yield logical Paulis:

¯ X1 = X1X2 = E(110000, 000000) ¯ Z1 = Z2Z6 = E(000000, 010001) ¯ X2 = X1X3 = E(101000, 000000) ¯ Z2 = Z3Z6 = E(000000, 001001)

2 Cliff26 ∼

= Sp(12, F2): CZ12E(a, b)CZ

† 12 = ±E

  • [a, b]FCZ12

. Find FCZ12. ¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001].

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 13 / 21

slide-32
SLIDE 32

Synthesizing Logical CZ12 for the [ [6, 4, 2] ] Code

Definition (partial) : CZ12 ¯ X1CZ

† 12 ¯

X1 ¯ Z2, CZ12 ¯ X2CZ

† 12 ¯

Z1 ¯ X2. The process:

1 Generator matrices for the [6, 5, 2] SPC code yield logical Paulis:

¯ X1 = X1X2 = E(110000, 000000) ¯ Z1 = Z2Z6 = E(000000, 010001) ¯ X2 = X1X3 = E(101000, 000000) ¯ Z2 = Z3Z6 = E(000000, 001001)

2 Cliff26 ∼

= Sp(12, F2): CZ12E(a, b)CZ

† 12 = ±E

  • [a, b]FCZ12

. Find FCZ12. ¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001].

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 13 / 21

slide-33
SLIDE 33

Synthesizing Logical CZ12 for the [ [6, 4, 2] ] Code

Definition (partial) : CZ12 ¯ X1CZ

† 12 ¯

X1 ¯ Z2, CZ12 ¯ X2CZ

† 12 ¯

Z1 ¯ X2. The process:

1 Generator matrices for the [6, 5, 2] SPC code yield logical Paulis:

¯ X1 = X1X2 = E(110000, 000000) ¯ Z1 = Z2Z6 = E(000000, 010001) ¯ X2 = X1X3 = E(101000, 000000) ¯ Z2 = Z3Z6 = E(000000, 001001)

2 Cliff26 ∼

= Sp(12, F2): CZ12E(a, b)CZ

† 12 = ±E

  • [a, b]FCZ12

. Find FCZ12. ¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001].

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 13 / 21

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SLIDE 34

Finding CZ12 via Sp(2m = 12, F2)

¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001] . . . . . . . . . . . .

(Also include constraints to normalize the stabilizer S) One possible solution

⇒ FCZ12 = I6 B I6

  • , B =

        1 1 1 1 1 1         ← →

⇒

CZ12= diag

  • ιvBv T

Z6 = CZ36CZ26CZ23Z6

Z 6 3 2

Not captured in FCZ12 – added to fix signs

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 14 / 21

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SLIDE 35

Finding CZ12 via Sp(2m = 12, F2)

¯ X1 = X1X2

CZ12

− → X1X2Z3Z6

γ,φ

⇐ ⇒ [110000, 000000]FCZ12 = [110000, 001001] ¯ X2 = X1X3

CZ12

− → X1X3Z2Z6

γ,φ

⇐ ⇒ [101000, 000000]FCZ12 = [101000, 010001] . . . . . . . . . . . .

(Also include constraints to normalize the stabilizer S) One possible solution

⇒ FCZ12 = I6 B I6

  • , B =

        1 1 1 1 1 1         ← →

⇒

CZ12= diag

  • ιvBv T

Z6 = CZ36CZ26CZ23Z6

Z 6 3 2

Not captured in FCZ12 – added to fix signs

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 14 / 21

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SLIDE 36

Algorithm to Synthesize Logical Clifford Operators

1 Determine the target ¯

g by specifying ¯ g ¯ Xi ¯ g† = ¯ X ′

i , ¯

g ¯ Zi ¯ g† = ¯ Z ′

i . Add

conditions to normalize or centralize S [Got09].

2 Using the maps γ, φ, transform these relations into linear equations

  • n F¯

g ∈ Sp(2m, F2), i.e., ¯

gE(a, b)¯ g † = ±E ([a, b]F¯

g) ⇒ [a, b] → [a, b]F¯ g.

3 Find the feasible symplectic solution set F¯

g using transvections.

4 Factor each F¯

g ∈ F¯ g using the decomposition in [Can17], and

compute the physical Clifford operator ¯ g.

5 Check for conjugation of ¯

g with S, ¯ Xi, ¯

  • Zi. If some signs are incorrect,

post-multiply ¯ g by an element from HWN as necessary (apply [NC10,

  • Prop. 10.4] to S⊥ = S, ¯

Xi, ¯ Zi).

6 Express ¯

g as a sequence of physical Clifford gates obtained from the factorization in step 4.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 15 / 21

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SLIDE 37

Symplectic Transvections

Definition: Given a vector h ∈ F2m

2 , the transvection Zh : F2m 2

→ F2m

2

is Zh(x) x + x, hsh ⇔ Fh = I2m + ΩhTh ∈ Sp(2m, F2). Fact: Transvections generate the binary symplectic group Sp(2m, F2). Lemma ([SAF08; KS14]) Let x, y ∈ F2m

2 . Then there exists at most two transvections Fh1, Fh2 s.t.

xFh1Fh2 = y. We extend this to map a sequence of vectors xi to yi, i = 1, . . . , t.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 16 / 21

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Our Results

Given a stabilizer code with logical Paulis ¯ Xi, ¯ Zi, we have the system   γ( ¯ X) γ(S) γ( ¯ Z)   F =   γ( ¯ X ′) γ(S′) γ( ¯ Z ′)   . Algorithm 1: Use symplectic transvections to find symplectic F s.t. xiF = yi, i = 1, . . . , t. Algorithm 2: Use “nullspace-like” ideas for symplectic matrices to enumerate all symplectic solutions F. Theorem: For an [ [m, m − k] ] stabilizer code, the number of symplectic solutions for each logical Clifford operator is 2k(k+1)/2. Theorem: For each logical Clifford operator of an [ [m, m − k] ] stabilizer code, one can always synthesize a solution that centralizes the stabilizer S.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 17 / 21

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Future Work

How to leverage this efficient enumeration during the process of computation? E.g., Quantum Compilers. What does this enumeration mean for topological codes? Understand the geometry of the solution space of symplectic matrices. Optimization of solutions with respect to a useful metric. Decomposition of symplectic matrix motivated by practical constraints, e.g., circuit complexity, fault-tolerance. Extend the framework to accommodate non-Clifford gates, e.g., T. . . . etc.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 18 / 21

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References

[Got97] Daniel Gottesman. “Stabilizer codes and quantum error correction”. PhD thesis. California Institute of Technology, 1997. [DD03] Jeroen Dehaene and Bart De Moor. “Clifford group, stabilizer states, and linear and quadratic operations over GF(2)”. In:

  • Phys. Rev. A 68.4 (Oct. 2003), p. 042318. doi:

10.1103/PhysRevA.68.042318. [SAF08] A Salam, E Al-Aidarous, and A El Farouk. “Optimal symplectic Householder transformations for SR decomposition”. In: Lin. Algebra and its Appl. 429 (2008), pp. 1334–1353. doi: 10.1016/j.laa.2008.02.029. [Got09] Daniel Gottesman. “An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation”. In: arXiv preprint arXiv:0904.2557 (2009).

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 19 / 21

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References

[Wil09] Mark M Wilde. “Logical operators of quantum codes”. In: Phys.

  • Rev. A 79.6 (2009), pp. 062322-1–062322-5. doi:

10.1103/PhysRevA.79.062322. [NC10] Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. [KS14] Robert Koenig and John A Smolin. “How to efficiently select an arbitrary Clifford group element”. In: J. Math. Phys. 55.12 (Dec. 2014), p. 122202. doi: 10.1063/1.4903507. [Can17] Trung Can. “An Algorithm to Generate a Unitary Transformation from Logarithmically Many Random Bits”. Research Independent Study Report, Duke Univeristy. 2017.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 20 / 21

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Thank you!

For details see https://arxiv.org/abs/1803.06987. Have fun synthesizing Clifford circuits for your favorite stabilizer code, at https://github.com/nrenga/symplectic-arxiv18a :-). Any feedback is much appreciated.

Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 21 / 21