baby's first diagrammatic calculus for quantum information...
TRANSCRIPT
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Baby's First Diagrammatic Calculus for Quantum Information
Processing
Vladimir Zamdzhiev
Department of Computer Science
Tulane University
30 May 2018
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Quantum computing
� Quantum computing is usually described using �nite-dimensional Hilbert spacesand linear maps (or �nite-dimensional C*-algebras and completely positive maps).
� Computing the matrix representation of quantum operations requires memoryexponential in the number of input qubits.
� This is not a scalable approach for software applications related to quantuminformation processing (QIP).
� An alternative is provided by the ZX-calculus which is a sound, complete anduniversal diagrammatic calculus for equational reasoning about �nite-dimensionalquantum computing.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Quantum computing
� Quantum computing is usually described using �nite-dimensional Hilbert spacesand linear maps (or �nite-dimensional C*-algebras and completely positive maps).
� Computing the matrix representation of quantum operations requires memoryexponential in the number of input qubits.
� This is not a scalable approach for software applications related to quantuminformation processing (QIP).
� An alternative is provided by the ZX-calculus which is a sound, complete anduniversal diagrammatic calculus for equational reasoning about �nite-dimensionalquantum computing.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
ZX-calculus
� The calculus is diagrammatic (some similarities to quantum circuits).� Example: Preparation of a Bell state.
H7→
|0〉
|0〉
� The ZX-calculus is practical. Used to study and discover new results in:� Quantum error-correcting codes [Chancellor, Kissinger, et. al 2016].� Measurement-based quantum computing [Duncan & Perdrix 2010].� (Quantum) foundations [Backens & Duman 2014].� and others...
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
ZX-calculus
� The ZX-calculus is formal.� Developed through the study of categorical quantum mechanics.� Rewrite system based on string diagrams of dagger compact closed categories.� Universal: Any linear map in FdHilb is the interpretation of some ZX-diagram D.� Sound: If ZX ` D1 = D2, then JD1K = JD2K.� Complete: If JD1K = JD2K, then ZX ` D1 = D2.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
ZX-calculus� The ZX-calculus is amenable to automation and formal reasoning.
� Implemented in the Quantomatic proof assistant.
Figure: The quantomatic proof assistant.5 / 38
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
ZX-calculus
� The ZX-calculus provides a di�erent conceptual perspective of quantuminformation.
Example
The Bell state is the standard example of an entangled state:
H7→
|0〉
|0〉= · · · =
The diagrammatic notation clearly indicates this is not a separable state.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
SyntaxA ZX-diagram is an open undirected graph constructed from the following generators:
α , ,α ,
where α ∈ [0, 2π).
Example
π π/2
π/4
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Syntax
The ZX-calculus is an equational theory. Equality is written as D1 = D2:
Example
=
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Normalisation
Remark: I ignore scalars and normalisation throughout the rest of the talk for brevity.But that can be handled by the language.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Semantics: wires
J K :=(1 00 1
)t |
:=
1 0 0 00 0 1 00 1 0 00 0 0 1
t |
= |00〉+ |11〉
t |
= 〈00|+ 〈11|
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Semantics: spiders and hadamard
J K =1√2
(1 11 −1
)= H
u
wwwv
α
}
���~
=
|0m〉 7→ |0n〉|1m〉 7→ e iα|1n〉others 7→ 0
u
wwwv
α
}
���~
=
|+m〉 7→ |+n〉|−m〉 7→ e iα|−n〉
others 7→ 0
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Semantics: tensors
If
u
v D1
}
~ = M1 and
u
v D2
}
~ = M2, then:
u
wwwwwwwwv
D1
D2
}
��������~
= M1 ⊗M2
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Semantics: composition
If
u
v D1
}
~ = M1 and
u
v D2
}
~ = M2, then:
u
v D1 D2
}
~ = M2 ◦M1
By following these rules we can represent any linear map f : C2m 7→ Cn as aZX-diagram (universality).
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: Quantum States
State ZX-diagram
|0〉|1〉 π
|+〉|−〉 π
|00〉+ |11〉
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: unitary operations
Unitary map ZX-diagram
Z π
X π
H
Z ◦ X ππ
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: 2-qubit gates
Unitary map ZX-diagram
Z ⊗ Xπ
π
∧Z
CNOT
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Proof systemThere are only 12 axioms. Five of them are:
α
β
= α+ β = =
=
; ; ;
; α α=
Remark: The color-swapped versions follow as derived rules.
Remark: This rewrite system is sound and complete, i.e. no need for linear algebra:
ZX ` D1 = D2 ⇐⇒ JD1K = JD2K.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: Preparation of Bell state
H7→
|0〉
|0〉= = =
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: CNOT is self-adjoint
Lemma: =
Then:
= == =7→
=
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
From Hilbert spaces to C*-algebras
� So far we talked about pure state quantum mechanics (FdHilb).
� Next, we show how to model mixed-states (FdCStar).
� I omit some details for simplicity (see [Coecke & Kissinger, Picturing QuantumProcesses]).
� The basic idea is to double up our diagrams and negate all angles in one of thecopies (but there are other ways as well).
Example
π/4 π π/2 7→π/4 π π/2
−π/4 π −π/2
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: Quantum States
State (FdHilb) ZX-diagram
|0〉|1〉 π
|+〉|−〉 π
|00〉+ |11〉
State (FdCStar) ZX-diagram
|0〉〈0|
|1〉〈1| π
π
|+〉〈+|
|−〉〈−| π
π
|00〉〈00|+ |11〉〈11|
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: unitary operations
Unitary map (FdHilb) ZX-diagram
Z π
X π
T π/4
H
Z ◦ X ππ
Unitary map (FdCStar) ZX-diagram
Zπ
π
Xπ
π
Tπ/4
−π/4
H
Z ◦ Xππ
π π
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: conditional unitary operations
Conditional unitary ZX-diagram
Zb
X b
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Measuremens
Measurement ZX-diagram
Measurement in Z basis
Measurement in X basis
Measurement in Bell basis
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
0. A qubit owned by Alice.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
1. Prepare Bell state.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
2. Do Bell basis measurement on both of Alice's qubits.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
3. Alice sends two bits (b1, b2) to Bob to inform him of measurement outcome.
b1
b2
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
4. Bob performs unitary correction X b1 ◦ Zb2 .
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
We can now prove the correctness of the teleportation protocol in the ZX-calculus:
=
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportationInput: A qubit owned by Alice.
==
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportationInput: A qubit owned by Alice.
==
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportationInput: A qubit owned by Alice.
==
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportationInput: A qubit owned by Alice.
==
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportationInput: A qubit owned by Alice.
==
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Example: quantum teleportation
=
Therefore, teleportation works as expected.
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Conclusion
� The ZX-calculus is a sound and complete alternative to linear algebra for�nite-dimensional quantum information processing.
� Great potential for formal methods, veri�cation and computer-aided reasoning.� Useful tool for studying QIP.� If you want to learn more, check out the book (contains outdated and incompleteversion of ZX):
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Introduction Syntax Semantics Proof System Pure states → mixed states Conclusion
Thank you for your attention!
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