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
Overview Motivation and our Contribution 1 Essential Algebraic Setup 2 Synthesis of Logical Clifford Operators for Stabilizer Codes 3 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 2 / 21
Encoded Computation: An Example Coded: [4 , 3 , 2] SPC Uncoded: 3 bits m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 m 1 ⊕ m 2 ⊕ m 3 1 1 0 1 1 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21
Encoded Computation: An Example Coded: [4 , 3 , 2] SPC Uncoded: 3 bits m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 X m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 m 1 ⊕ m 2 ⊕ m 3 1 1 0 1 1 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21
Encoded Computation: An Example Coded: [4 , 3 , 2] SPC Uncoded: 3 bits m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 X m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 m 1 ⊕ m 2 ⊕ m 3 1 1 0 1 1 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21
Encoded Computation: An Example Uncoded: 3 bits Coded: [4 , 3 , 2] SPC m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 X m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 1 1 0 m 1 ⊕ m 3 ⊕ 1 1 1 KEY: Physical circuit has to realize logical operation & preserve the code. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21
Encoded Computation: An Example Uncoded: 3 bits Coded: [4 , 3 , 2] SPC m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 X X m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 1 1 0 m 1 ⊕ m 3 ⊕ 1 1 1 KEY: Physical circuit has to realize logical operation & preserve the code. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 3 / 21
Encoded Computation: An Example Uncoded: 3 bits Coded: [4 , 3 , 2] SPC m 1 m 1 ⊕ m 2 m 1 m 1 ⊕ m 2 X X m 2 m 2 m 2 m 2 m 3 m 3 ⊕ 1 m 3 m 3 ⊕ 1 1 1 m 1 ⊕ m 3 ⊕ 1 0 1 1 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
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
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
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
Problem: Operations on Encoded Qubits unreliable circuit arbitrary logical operation initial desired | x � L | ˜ x � L final logical state state QECC QECC encode decode relevant physical operation | ψ x � | ψ ˜ x � QECC: Quantum Error-Correcting Codes Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 5 / 21
Problem: Operations on Encoded Qubits unreliable circuit arbitrary logical operation initial desired | x � L | ˜ x � L final logical state state Need to translate QECC QECC for given encode decode QECC relevant physical operation | ψ x � | ψ ˜ x � 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
Algorithms on GitHub Our algorithms are available open-source at: https://github.com/nrenga/symplectic-arxiv18a Logical Clifford Our All physical circuits ¯ g that realize g L & fix S Operator g L Algorithm Logical Paulis ¯ X i , ¯ Stabilizer S Z i (defines the code) [Got97; Wil09] Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 6 / 21
Overview Motivation and our Contribution 1 Essential Algebraic Setup 2 Synthesis of Logical Clifford Operators for Stabilizer Codes 3 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 7 / 21
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. 0 � 1 � � 0 � 1 Example ( m = 2 qubits) : | 0 � ⊗ | 1 � = ⊗ = = | 01 � . 0 1 0 0 m Qubits: If qubit i is in the state | v i � ∈ {| 0 � , | 1 �} then the Kronecker product | v 1 � ⊗ · · · ⊗ | v m � � | v � describes the state of the system. Note that C N = C 2 m = C 2 ⊗ · · · ⊗ C 2 ( m times). N = 2 m . 2 | α v | 2 = 1. Pure state ( m qubits): | φ � = � 2 α v | v � , α v ∈ C , � v ∈ F m v ∈ F m Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 8 / 21
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. 0 � 1 � � 0 � 1 Example ( m = 2 qubits) : | 0 � ⊗ | 1 � = ⊗ = = | 01 � . 0 1 0 0 m Qubits: If qubit i is in the state | v i � ∈ {| 0 � , | 1 �} then the Kronecker product | v 1 � ⊗ · · · ⊗ | v m � � | v � describes the state of the system. Note that C N = C 2 m = C 2 ⊗ · · · ⊗ C 2 ( m times). N = 2 m . 2 | α v | 2 = 1. Pure state ( m qubits): | φ � = � 2 α v | v � , α v ∈ C , � v ∈ F m v ∈ F m Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 8 / 21
Heisenberg-Weyl Group HW N The Heisenberg-Weyl (or Pauli) group for a single qubit: √ HW 2 � ι κ { I 2 , X , Z , Y } , ι � − 1 , κ ∈ { 0 , 1 , 2 , 3 } . � 0 � 1 X � Bit-Flip: ⇒ X | 0 � = | 1 � , X | 1 � = | 0 � . 1 0 � 1 � 0 Z � Phase-Flip: ⇒ Z | 0 � = | 0 � , Z | 1 � = − | 1 � . 0 − 1 � 0 � − ι Y � Bit-Phase Flip: = ι XZ . XZ = − ZX . 0 ι Fact: { I 2 , X , Z , Y } forms an orthonormal basis for operators in C 2 × 2 . For m Qubits: HW N � Kronecker products of m HW 2 matrices ( N = 2 m ). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21
Heisenberg-Weyl Group HW N The Heisenberg-Weyl (or Pauli) group for a single qubit: √ HW 2 � ι κ { I 2 , X , Z , Y } , ι � − 1 , κ ∈ { 0 , 1 , 2 , 3 } . � 0 � 1 X � Bit-Flip: ⇒ X | 0 � = | 1 � , X | 1 � = | 0 � . 1 0 � 1 � 0 Z � Phase-Flip: ⇒ Z | 0 � = | 0 � , Z | 1 � = − | 1 � . 0 − 1 � 0 � − ι Y � Bit-Phase Flip: = ι XZ . XZ = − ZX . 0 ι Fact: { I 2 , X , Z , Y } forms an orthonormal basis for operators in C 2 × 2 . For m Qubits: HW N � Kronecker products of m HW 2 matrices ( N = 2 m ). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arXiv:1803.06987 9 / 21
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