kerdock codes determine unitary 2-designs
Post on 02-Feb-2022
2 Views
Preview:
TRANSCRIPT
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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 2 / 14
Overview
1 Motivation and our Contributions
2 Essential Algebraic Setup
3 Stabilizer States and Kerdock Codes
4 Kerdock Codes Determine Unitary 2-Designs
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 2 / 14
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 ) =
∫UN
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.
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 3 / 14
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 ) =
∫UN
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.
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 3 / 14
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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 4 / 14
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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 4 / 14
Overview
1 Motivation and our Contributions
2 Essential Algebraic Setup
3 Stabilizer States and Kerdock Codes
4 Kerdock Codes Determine Unitary 2-Designs
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 4 / 14
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
.
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 5 / 14
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
.
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 5 / 14
Clifford Group and Symplectic Matrices
Binary Representation: X ⊗ Z ⊗ Y = E ( 1 0 1, 0 1 1 ) = E (a, b), a, b ∈ Fm2 .
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))!
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 6 / 14
Clifford Group and Symplectic Matrices
Binary Representation: X ⊗ Z ⊗ Y = E ( 1 0 1, 0 1 1 ) = E (a, b), a, b ∈ Fm2 .
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))!
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 6 / 14
Elementary Symplectic Matrices
Symplectic Matrix Fg Physical Operator g Clifford Element
Ω =
[0 ImIm 0
]HN = H⊗m2 = 1√
2m
[1 11 −1
]⊗mTransversal
Hadamard
AQ =
[Q 00 Q−T
]aQ =
∑v∈Fm
2|vQ〉 〈v | CNOTs,
Permutations
TP =
[Im P0 Im
]with P symmetric
tP =∑
v∈Fm2ıvPv
T mod 4 |v〉 〈v | Phase Gates,Controlled-Z (CZ)
Gk =
[Lm−k Uk
Uk Lm−k
]Uk = diag (Ik ,Om−k)
Lm−k = diag (Ok , Im−k)
gk = H2k ⊗ I2m−kPartial
Hadamards
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 7 / 14
Elementary Symplectic Matrices
Symplectic Matrix Fg Physical Operator g Clifford Element
Ω =
[0 ImIm 0
]HN = H⊗m2 = 1√
2m
[1 11 −1
]⊗mTransversal
Hadamard
AQ =
[Q 00 Q−T
]aQ =
∑v∈Fm
2|vQ〉 〈v | CNOTs,
Permutations
TP =
[Im P0 Im
]with P symmetric
tP =∑
v∈Fm2ıvPv
T mod 4 |v〉 〈v | Phase Gates,Controlled-Z (CZ)
Gk =
[Lm−k Uk
Uk Lm−k
]Uk = diag (Ik ,Om−k)
Lm−k = diag (Ok , Im−k)
gk = H2k ⊗ I2m−kPartial
Hadamards
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 7 / 14
Stabilizer States (SS)
Paulis: E (a, b) =(ıa1b1X a1Zb1
)⊗(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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 8 / 14
Stabilizer States (SS)
Paulis: E (a, b) =(ıa1b1X a1Zb1
)⊗(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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 8 / 14
Stabilizer States (SS)
Paulis: E (a, b) =(ıa1b1X a1Zb1
)⊗(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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 8 / 14
Stabilizer States (SS)
Paulis: E (a, b) =(ıa1b1X a1Zb1
)⊗(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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 8 / 14
Overview
1 Motivation and our Contributions
2 Essential Algebraic Setup
3 Stabilizer States and Kerdock Codes
4 Kerdock Codes Determine Unitary 2-Designs
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 8 / 14
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
2
[(vPvT + 2vwT ) mod 4
]|v〉 ∈ ZN
4
Binary Kerdock Code
Gray Map: ZN4 → F2N
2
Squared Euclidean Distance= 2 × Hamming Distance
Z4-Linear Kerdock Code
Exponentiation!
Orthonormal Basisof Stabilizer States
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
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
2
[(vPvT + 2vwT ) mod 4
]|v〉 ∈ ZN
4
Binary Kerdock Code
Gray Map: ZN4 → F2N
2
Squared Euclidean Distance= 2 × Hamming Distance
Z4-Linear Kerdock Code
Exponentiation!
Orthonormal Basisof Stabilizer States
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
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
2
[(vPvT + 2vwT ) mod 4
]|v〉 ∈ ZN
4
Binary Kerdock Code
Gray Map: ZN4 → F2N
2
Squared Euclidean Distance= 2 × Hamming Distance
Z4-Linear Kerdock Code
Exponentiation!
Orthonormal Basisof Stabilizer States
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
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
2
[(vPvT + 2vwT ) mod 4
]|v〉 ∈ ZN
4
Binary Kerdock Code
Gray Map: ZN4 → F2N
2
Squared Euclidean Distance= 2 × Hamming Distance
Z4-Linear Kerdock Code
Exponentiation!
Orthonormal Basisof Stabilizer States
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
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
2
[(vPvT + 2vwT ) mod 4
]|v〉 ∈ ZN
4
Binary Kerdock Code
Gray Map: ZN4 → F2N
2
Squared Euclidean Distance= 2 × Hamming Distance
Z4-Linear Kerdock Code
Exponentiation!
Orthonormal Basisof Stabilizer States
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
Overview
1 Motivation and our Contributions
2 Essential Algebraic Setup
3 Stabilizer States and Kerdock Codes
4 Kerdock Codes Determine Unitary 2-Designs
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 9 / 14
Clifford Symmetries of Kerdock SSs
gP,w =(∑
v∈Fm2ıvPv
T |v〉 〈v |)· HN · E (w , 0)⇒ gP,w |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
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
Ω =
[0 ImIm 0
]TP =
[Im P0 Im
]Kerdock Set
of Matrices P:P 6= Q ⇒ P + Q
non-singularAssociate
z ∈ F2m ↔ Pz
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 10 / 14
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 .
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 11 / 14
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!
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 12 / 14
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
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 13 / 14
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?
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 14 / 14
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 appreciated!narayanan.rengaswamy@duke.edu
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 14 / 14
Weight Distribution of (Binary) Kerdock Codes
Kerdock codewords:∑
v∈Fm2
[(vPvT + 2vwT + κ) mod 4
]|v〉 ∈ ZN
4
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 ]).
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 14 / 14
Weight Distribution of (Binary) Kerdock Codes
Kerdock codewords:∑
v∈Fm2
[(vPvT + 2vwT + κ) mod 4
]|v〉 ∈ ZN
4
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 ]).
Narayanan Rengaswamy Kerdock Codes Determine Unitary 2-Designs arXiv:1904.07842, 1907.00310 14 / 14
top related