Kerdock Codes Determine Unitary 2-Designs

Narayanan RengaswamyInformation Initiative at Duke (iiD), Duke University

Joint Work: Trung Can, Robert Calderbank, and Henry Pfister

2019 IEEE Intl. Symposium on Information TheoryParis, France

arXiv: 1904.07842, 1907.00310

July 12, 2019

Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 1 / 14

Overview

1 Motivation and our Contributions

2 Essential Algebraic Setup

3 Stabilizer States and Kerdock Codes

4 Kerdock Codes Determine Unitary 2-Designs

Overview

Problem and Motivation

Randomized Benchmarking: Procedure to estimate the quality ofgates on a quantum computer.

Twirl the noise channel through a randomized sequence of gates andinduce a depolarizing channel.

Noise fidelity is invariant under twirling, so suffices to estimate thefidelity of the depolarizing channel.

Unitary 2-design: The gates must be chosen from an ensemble ofunitary matrices E = pi ,Uiti=1 such that

t∑i=1

pi (Ui ⊗ Ui )X (U†i ⊗ U†i ) =

dµ (U ⊗ U)X (U† ⊗ U†),

where X is any linear operator on CN ⊗ CN and dµ is the Haarmeasure on UN , the unitary group on m = log2(N) qubits.

Problem and Motivation

Randomized Benchmarking: Procedure to estimate the quality ofgates on a quantum computer.

Twirl the noise channel through a randomized sequence of gates andinduce a depolarizing channel.

Noise fidelity is invariant under twirling, so suffices to estimate thefidelity of the depolarizing channel.

Unitary 2-design: The gates must be chosen from an ensemble ofunitary matrices E = pi ,Uiti=1 such that

t∑i=1

pi (Ui ⊗ Ui )X (U†i ⊗ U†i ) =

dµ (U ⊗ U)X (U† ⊗ U†),

where X is any linear operator on CN ⊗ CN and dµ is the Haarmeasure on UN , the unitary group on m = log2(N) qubits.

Main Contributions

Implementation Online:https://github.com/nrenga/symplectic-arxiv18a

The permutation automorphism group of the Z4-linear Kerdock codesproduces a unitary 2-design, of almost optimal size.

A simple derivation of the weight distribution of Kerdock codes.

Key Classical-Quantum Connection:

Exponentiated Kerdock codewords are stabilizer states.

Hamming Distance ←→ Inner Products

Permutations ←→ Clifford Symmetries

Main Contributions

Implementation Online:https://github.com/nrenga/symplectic-arxiv18a

The permutation automorphism group of the Z4-linear Kerdock codesproduces a unitary 2-design, of almost optimal size.

A simple derivation of the weight distribution of Kerdock codes.

Key Classical-Quantum Connection:

Exponentiated Kerdock codewords are stabilizer states.

Hamming Distance ←→ Inner Products

Permutations ←→ Clifford Symmetries

Overview

Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit:

HW2 , 〈ıκI2,X ,Z ,Y 〉, ı ,√−1, κ ∈ Z4, I2,X ,Y ,Z ∈ C2×2.

Bit-Flip : X |0〉 = |1〉 , X |1〉 = |0〉 .Phase-Flip : Z |0〉 = |0〉 , Z |1〉 = − |1〉 .

Bit-Phase Flip : Y , ı · XZ ⇒ Y |x〉 = ı · (−1)x |x ⊕ 1〉 .

For m Qubits: HWN , Kronecker products of m HW2 matrices (N = 2m).

Binary Representation: X ⊗ Z ⊗ Y = E ( 1 0 1, 0 1 1 ) = E (a, b), a, b ∈ Fm2 .

XZ = −ZX : E (a, b),E (c , d) commute iff adT + bcT = 0︸︷︷︸symplectic inner product

Heisenberg-Weyl Group HWN

The Heisenberg-Weyl (or Pauli) group for a single qubit:

HW2 , 〈ıκI2,X ,Z ,Y 〉, ı ,√−1, κ ∈ Z4, I2,X ,Y ,Z ∈ C2×2.

Bit-Flip : X |0〉 = |1〉 , X |1〉 = |0〉 .Phase-Flip : Z |0〉 = |0〉 , Z |1〉 = − |1〉 .

Bit-Phase Flip : Y , ı · XZ ⇒ Y |x〉 = ı · (−1)x |x ⊕ 1〉 .

For m Qubits: HWN , Kronecker products of m HW2 matrices (N = 2m).

XZ = −ZX : E (a, b),E (c , d) commute iff adT + bcT = 0︸︷︷︸symplectic inner product

Clifford Group and Symplectic Matrices

Paulis: E (a, b) =(ıa1b1X a1Zb1

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Clifford Group: All unitaries that map Paulis to Paulis under conjugation.

Symplectic Matrices: If g ∈ CliffN (Cliffords on m = log2N qubits) then

g E (a, b) g † = ±E ([a, b]Fg ) , where FgΩFTg = Ω =

[0 ImIm 0

Fg ∈ F2m×2m2 is symplectic: preserves symplectic inner products.

The Clifford group is a unitary 2-design but is too big (2O(m2))!

Clifford Group and Symplectic Matrices

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Clifford Group: All unitaries that map Paulis to Paulis under conjugation.

Symplectic Matrices: If g ∈ CliffN (Cliffords on m = log2N qubits) then

g E (a, b) g † = ±E ([a, b]Fg ) , where FgΩFTg = Ω =

[0 ImIm 0

Fg ∈ F2m×2m2 is symplectic: preserves symplectic inner products.

The Clifford group is a unitary 2-design but is too big (2O(m2))!

Elementary Symplectic Matrices

Symplectic Matrix Fg Physical Operator g Clifford Element

[0 ImIm 0

]HN = H⊗m2 = 1√

[1 11 −1

]⊗mTransversal

Hadamard

[Q 00 Q−T

∑v∈Fm

2|vQ〉〈v | CNOTs,

Permutations

[Im P0 Im

]with P symmetric

tP =∑

v∈Fm2ıvPv

T mod 4 |v〉〈v | Phase Gates,Controlled-Z (CZ)

[Lm−k Uk

Uk Lm−k

]Uk = diag (Ik ,Om−k)

Lm−k = diag (Ok , Im−k)

gk = H2k ⊗ I2m−kPartial

Hadamards

Elementary Symplectic Matrices

Symplectic Matrix Fg Physical Operator g Clifford Element

[0 ImIm 0

]HN = H⊗m2 = 1√

[1 11 −1

]⊗mTransversal

Hadamard

[Q 00 Q−T

∑v∈Fm

2|vQ〉〈v | CNOTs,

Permutations

[Im P0 Im

]with P symmetric

tP =∑

v∈Fm2ıvPv

T mod 4 |v〉〈v | Phase Gates,Controlled-Z (CZ)

[Lm−k Uk

Uk Lm−k

]Uk = diag (Ik ,Om−k)

Lm−k = diag (Ok , Im−k)

gk = H2k ⊗ I2m−kPartial

Hadamards

Stabilizer States (SS)

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Cliffords: If g ∈ CliffN , then g E (a, b) g † = ±E ([a, b]Fg ), Fg symplectic.

Stabilizer: Commutative subgroup of the Pauli group HWN .

SS: The common eigenvectors of maximal (size) stabilizers.

Z |0〉 = |0〉 ⇒ E (0, b) |0〉⊗m = |0〉⊗m ⇒ ±E ([0, b]Fg ) · g |0〉⊗m = g |0〉⊗m.

SS g |0〉⊗m ←→ maximal stabilizer ±E ([0, b]Fg ), b ∈ Fm2 .

Example:

g =(∑

v∈Fm2ıvPv

T |v〉〈v |)· HN · E (w , 0)⇒ g |0〉⊗m ∝

∑v∈Fm

2ı(vPv

T+2vwT ) mod 4 |v〉

(P = PT ∈ Fm×m2 , w ∈ Fm

2 ) E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Z |0〉 = |0〉 ⇒ E (0, b) |0〉⊗m = |0〉⊗m ⇒ ±E ([0, b]Fg ) · g |0〉⊗m = g |0〉⊗m.

Example:

g =(∑

v∈Fm2ıvPv

T |v〉〈v |)· HN · E (w , 0)⇒ g |0〉⊗m ∝

∑v∈Fm

2ı(vPv

(P = PT ∈ Fm×m2 , w ∈ Fm

2 ) E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Z |0〉 = |0〉 ⇒ E (0, b) |0〉⊗m = |0〉⊗m ⇒ ±E ([0, b]Fg ) · g |0〉⊗m = g |0〉⊗m.

Example:

g =(∑

v∈Fm2ıvPv

T |v〉〈v |)· HN · E (w , 0)⇒ g |0〉⊗m ∝

∑v∈Fm

2ı(vPv

(P = PT ∈ Fm×m2 , w ∈ Fm

2 ) E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector

)⊗(ıa2b2X a2Zb2

)⊗ · · · ⊗

(ıambmX amZbm

Z |0〉 = |0〉 ⇒ E (0, b) |0〉⊗m = |0〉⊗m ⇒ ±E ([0, b]Fg ) · g |0〉⊗m = g |0〉⊗m.

Example:

g =(∑

v∈Fm2ıvPv

T |v〉〈v |)· HN · E (w , 0)⇒ g |0〉⊗m ∝

∑v∈Fm

2ı(vPv

(P = PT ∈ Fm×m2 , w ∈ Fm

2 ) E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector

Overview

Connecting Quantum and Classical Worlds

∑v∈Fm

2|v〉 ⊗

[g(vPvT + 2vwT )

]T ∈ F2N2

g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector∑v∈Fm

2ı(vPv

T+2vwT ) mod 4 |v〉 ∈ ±1,±ıN

exponentiation∑v∈Fm

[(vPvT + 2vwT ) mod 4

]|v〉 ∈ ZN

Binary Kerdock Code

Gray Map: ZN4 → F2N

Squared Euclidean Distance= 2 × Hamming Distance

Z4-Linear Kerdock Code

Exponentiation!

Orthonormal Basisof Stabilizer States

∑v∈Fm

2|v〉 ⊗

[g(vPvT + 2vwT )

]T ∈ F2N2

g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

E ([Im |P]) = ±E (b, bP), b ∈ Fm2

2ı(vPv

T+2vwT ) mod 4 |v〉 ∈ ±1,±ıN

]|v〉 ∈ ZN

Binary Kerdock Code

Exponentiation!

∑v∈Fm

2|v〉 ⊗

[g(vPvT + 2vwT )

]T ∈ F2N2

g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

E ([Im |P]) = ±E (b, bP), b ∈ Fm2

2ı(vPv

T+2vwT ) mod 4 |v〉 ∈ ±1,±ıN

]|v〉 ∈ ZN

Binary Kerdock Code

Exponentiation!

∑v∈Fm

2|v〉 ⊗

[g(vPvT + 2vwT )

]T ∈ F2N2

g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

E ([Im |P]) = ±E (b, bP), b ∈ Fm2

2ı(vPv

T+2vwT ) mod 4 |v〉 ∈ ±1,±ıN

]|v〉 ∈ ZN

Binary Kerdock Code

Exponentiation!

∑v∈Fm

2|v〉 ⊗

[g(vPvT + 2vwT )

]T ∈ F2N2

g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

E ([Im |P]) = ±E (b, bP), b ∈ Fm2

2ı(vPv

T+2vwT ) mod 4 |v〉 ∈ ±1,±ıN

]|v〉 ∈ ZN

Binary Kerdock Code

Exponentiation!

Overview

Clifford Symmetries of Kerdock SSs

gP,w =(∑

v∈Fm2ıvPv

T |v〉〈v |)· HN · E (w , 0)⇒ gP,w |0〉⊗m ∝

∑v∈Fm

2ı(vPv

(P = PT ∈ Fm×m2 , w ∈ Fm

2 ) E ([Im |P]) = ±E (b, bP), b ∈ Fm2

eigenvector

ZN = E ([0 | Im])

E (w , 0) |0〉⊗m = |w〉

XN = E ([Im | 0])

HN |w〉 ∝∑

v∈Fm2

(−1)vwT |v〉

YP = E ([Im |P])

tPHN |w〉 ∝∑

v∈Fm2ıvPv

T+2vwT |v〉

UnitaryOperator

HN tP =∑

v∈Fm2ıvPv

T |v〉〈v |

SymplecticMatrix

[0 ImIm 0

[Im P0 Im

]Kerdock Set

of Matrices P:P 6= Q ⇒ P + Q

non-singularAssociate

z ∈ F2m ↔ Pz

Kerdock Symmetries

Col. of Mz indexed by w ∈ Fm2 : |ψPz ,w 〉 ∝

∑v∈Fm

2ı(vPzvT+2vwT ) mod 4 |v〉.

Cols. of Mz form the eigenbasis of E ([Im |Pz ]) = ±E (b, bPz), b ∈ Fm2 .

Form the N × N(N + 1) matrix M , [ M∞ | M0 | · · · | Mz | · · · ].

Each of the N + 1 blocks of M correspond to a stabilizer E ([Im |Pz ]).

Symmetry of M: A pair (U,G ) s.t. UMG = M, where U ∈ UN andG is a generalized permutation matrix with entries in 1, ı,−1,−ı.

Lemma: For any symmetry (U,G ) of M, U ∈ CliffN .

Proof Idea: U permutes the stabilizers E ([Im |Pz ]), so U ∈ CliffN .

Kerdock Symmetries form a Unitary 2-Design

Col. of Mz indexed by w ∈ Fm2 : |ψPz ,w 〉 ∝

∑v∈Fm

2ı(vPzvT+2vwT ) mod 4 |v〉.

Cols. of Mz form the eigenbasis of E ([Im |Pz ]) = ±E (b, bPz), b ∈ Fm2 .

Form the N × N(N + 1) matrix M , [ M∞ | M0 | · · · | Mz | · · · ].

Pauli Mixing: Transitivity on Paulis, implies unitary 2-design (Webb).

Symmetry Group PK,m of M: Generated as a product of 3 subgroups,each of which is one-to-one with a generator of the projective speciallinear group PSL(2,N). (Use symplectic matrices for this connection.)

PK,m∼= PSL(2,N): Size (N + 1)N(N − 1) ≈ 23m |CliffN | ≈ 2O(m2).

PK,m is Pauli mixing and hence forms a unitary 2-design!

Logical Unitary 2-Designs

Combining with our Logical Clifford Synthesis (LCS) algorithm(arXiv:1907.00310), we can synthesize unitary 2-designs on the qubits

protected by a (quantum) stabilizer error-correcting code.

Code: https://github.com/nrenga/symplectic-arxiv18a

LCSAlgorithm

Stabilizer S(defines the code)

Logical Paulis Xi , Zi

(GaussianElimination)

Logical CliffordOperator gL

All physical circuits gthat realize gL & fix S

Summary and Future Work

Exponentiated Kerdock codewords are stabilizer states (SS).

Connection simplifies derivation of the Kerdock weight distribution.

Clifford symmetries of Kerdock SS form a small unitary 2-design.

The design is isomorphic to Cleve et al. (arXiv:1501.04592), but theclassical coding connection is new and makes the description simple.

The isomorphism to PSL(2,N) makes sampling from the design easy.

Using LCS algorithm, produced logical unitary 2-designs. Applicationin logical randomized benchmarking protocol (arXiv:1702.03688).

Make an approximate unitary 2-design with lower circuit complexity?

Use coding connection to synthesize unitary t-designs for t > 2?

Thank you!

For details see http://arxiv.org/abs/1904.07842

and http://arxiv.org/abs/1907.00310

(Logical/Physical) Unitary 2-Design Implementation:https://github.com/nrenga/symplectic-arxiv18a

Any feedback is much [email protected]

Weight Distribution of (Binary) Kerdock Codes

Kerdock codewords:∑

v∈Fm2

[(vPvT + 2vwT + κ) mod 4

]|v〉 ∈ ZN

Subtracting two codewords ←→ Inner product of corresponding SS!

Gray Map: Z4 → F22 g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

Lemma: For u1, u2 ∈ 1, ı,−1,−ıN ,

〈u1 − u2, u1 − u2〉 = 2dH(g(u1), g(u2))

Exponentiated Kerdock Codeword

Lemma: For P1,P2 ∈ PK(m), |〈u1, u2〉|2 =

0 if P1 = P2 and u1 6= u2

2m if P1 6= P2,

22m if (P1 = P2 and) u1 = u2.

Proof Idea: |〈u1, u2〉|2 = Tr[(u1u

†1)(u2u

†2)], uiu

†i = Projector onto ui ←→ E([Im |Pi ]).

Weight Distribution of (Binary) Kerdock Codes

Kerdock codewords:∑

v∈Fm2

[(vPvT + 2vwT + κ) mod 4

]|v〉 ∈ ZN

Subtracting two codewords ←→ Inner product of corresponding SS!

Gray Map: Z4 → F22 g(1/ı) = 01

g(3/− ı) = 10

g(0/1) = 00g(2/− 1) = 11

Lemma: For u1, u2 ∈ 1, ı,−1,−ıN ,

〈u1 − u2, u1 − u2〉 = 2dH(g(u1), g(u2))

Exponentiated Kerdock Codeword

Lemma: For P1,P2 ∈ PK(m), |〈u1, u2〉|2 =

0 if P1 = P2 and u1 6= u2

2m if P1 6= P2,

22m if (P1 = P2 and) u1 = u2.

Proof Idea: |〈u1, u2〉|2 = Tr[(u1u

†1)(u2u

†2)], uiu

†i = Projector onto ui ←→ E([Im |Pi ]).

Kerdock Codes Determine Unitary 2-Designs

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