Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State...

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Quantum Error Correction I Protecting Quantum Information Manny Qubits are subsystems. Error control methods. Algebraic error models. Error detection. Error correction. Stabilizer codes.

Transcript of Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State...

Page 1: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

Quantum Error Correction IProtecting Quantum Information

Manny

• Qubits are subsystems.

• Error control methods.

• Algebraic error models.

• Error detection.

• Error correction.

• Stabilizer codes.

Page 2: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

1LOS ALAMOSNational Laboratory

One of Two Qubits

A ⊂ A B

Page 3: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

1LOS ALAMOSNational Laboratory

One of Two Qubits

A ⊂ A B

• State spaces:α 0〉

A+ β 1〉

Bα 00〉

AB+ β 01〉

AB+ γ 10〉

AB+ δ 11〉

AB

Q ⊂? Q⊗Q

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1LOS ALAMOSNational Laboratory

One of Two Qubits

A ⊂ A B

• State spaces:α 0〉

A+ β 1〉

Bα 00〉

AB+ β 01〉

AB+ γ 10〉

AB+ δ 11〉

AB

Q ⊂? Q⊗Q

• Observable algebras:σx

(A), σy(A), σz

(A), . . . ⊂ σx(A), . . . , σx

(B), . . . , σx(A)σx

(B), . . .

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Passive Error Control

A B

• Example noise operators:I, σx

(B), σy(B), σz

(B).

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Passive Error Control

A B

• Example noise operators:I, σx

(B), σy(B), σz

(B).

Information in A is protected.

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Passive Error Control

A B

• Example noise operators:I, σx

(B), σy(B), σz

(B).

Information in A is protected.No state of AB is protected.

00〉AB

σx(B)

→ 01〉AB

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Passive Error Control

A B

• Example noise operators:I, σx

(B), σy(B), σz

(B).

Information in A is protected.No state of AB is protected.

00〉AB

σx(B)

→ 01〉AB

Observables for A commute with errors.σu

(A)σv(B) = σv

(B)σu(A).

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Active Error Control

A B

• Example noise operators:I, cxnot = σx

(B)cnot(BA).

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Active Error Control

A B

• Example noise operators:I, cxnot = σx

(B)cnot(BA).

Start with B in 0〉B.0〉

L= 00〉

AB, 1〉

L= 10〉

AB

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Active Error Control

A B

• Example noise operators:I, cxnot = σx

(B)cnot(BA).

Start with B in 0〉B.0〉

L= 00〉

AB, 1〉

L= 10〉

AB

Information of A preserved after one error.

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Active Error Control

A B

• Example noise operators:I, cxnot = σx

(B)cnot(BA).

Start with B in 0〉B.0〉

L= 00〉

AB, 1〉

L= 10〉

AB

Information of A preserved after one error.. . . lost after two.

0〉L

cxnot→ 01〉AB

cxnot→ 10〉AB

= 1〉L

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Active Error Control

A B

• Example noise operators:I, cxnot = σx

(B)cnot(BA).

Start with B in 0〉B.0〉

L= 00〉

AB, 1〉

L= 10〉

AB

Information of A preserved after one error.. . . lost after two.

0〉L

cxnot→ 01〉AB

cxnot→ 10〉AB

= 1〉L

Solution: Reset B before errors.

Back to: Error Correction I

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Error Control Methods

• Passive error control.Noiseless subsystems.

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Error Control Methods

• Passive error control.Noiseless subsystems.

• Error suppression.Refocusing.Active symmetryenforcement.Decoupling.Reservoir engineering.

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Error Control Methods

• Passive error control.Noiseless subsystems.

• Error suppression.Refocusing.Active symmetryenforcement.Decoupling.Reservoir engineering.

• Systematic error control.

Rotating frames.Composite pulses.Adiabatic gates.

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Error Control Methods

• Passive error control.Noiseless subsystems.

• Error suppression.Refocusing.Active symmetryenforcement.Decoupling.Reservoir engineering.

• Systematic error control.

Rotating frames.Composite pulses.Adiabatic gates.

• Active error control.Periodic error correction.

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References

• Prehistory:Quantum Zeno effect: Misra&Sudarshan 1977 [14].Deutsch 1993, Barenco&al. 1996 [3].

• Discovery and Theory:Shor 1995 [17], Steane 1995 [19].Bennett&DiVincenzo&Smolin&Wootters 1996 [4], Knill&Laflamme 1996 [10].Calderbank&Shor 1996 [6], Gottesman 1996 [7], Calderbank&Rains&Shor&Sloane1997 [5].

• Fault tolerance and threshold accuracies:Shor 1996 [18], Kitaev 1997 [9].Aharonov&Ben-Or 1996 [1, 2], Knill&Laflamme&Zurek 1996 [12],Gottesman&Preskill 1997 [8, 16].

• Toward subsystems:Quasi-particles . . .Zanardi&Rasetti 1997 [23], Lidar&Chuang&Whaley 1998 [13].Viola&Knill&Lloyd 1998 [21, 20, 22].Knill&Laflamme 1996 [10], Knill&Laflamme&Viola 2000 [11].

General reference: (M)ike, Ch. 10. Nielsen&Chuang 2001 [15]

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The Pauli Error Model

1 2 3 . . .• Error operators:

E1 ={I, σx

(1), σy(1), σz

(1), σx(2), σy

(2), . . .}

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The Pauli Error Model

1 2 3 . . .• Error operators:

E1 ={I, σx

(1), σy(1), σz

(1), σx(2), σy

(2), . . .}

• Weight 2 error operators:E2 = E1E1

={I, σx

(1), . . . σx(1)σx

(2), σx(1)σx

(3), . . .}

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The Pauli Error Model

1 2 3 . . .• Error operators:

E1 ={I, σx

(1), σy(1), σz

(1), σx(2), σy

(2), . . .}

• Weight 2 error operators:E2 = E1E1

={I, σx

(1), . . . σx(1)σx

(2), σx(1)σx

(3), . . .}

• Higher weights:

Ek = Ek1 =k times︷ ︸︸ ︷E1E1 . . .

E =⋃∞k=1 Ek

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The Pauli Error Model

1 2 3 . . .• Error operators:

E1 ={I, σx

(1), σy(1), σz

(1), σx(2), σy

(2), . . .}

• Weight 2 error operators:E2 = E1E1

={I, σx

(1), . . . σx(1)σx

(2), σx(1)σx

(3), . . .}

• Higher weights:

Ek = Ek1 =k times︷ ︸︸ ︷E1E1 . . .

E =⋃∞k=1 Ek

The linear span of E contains all operators.

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Algebraic Error Models

• Weight 1 error events:E1 = { I, E1, E2, . . .}

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Algebraic Error Models

• Weight 1 error events:E1 = { I, E1, E2, . . .}

• Weight k error events:

Ek = Ek1 =k times︷ ︸︸ ︷E1E1 . . .

Page 25: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Algebraic Error Models

• Weight 1 error events:E1 = { I, E1, E2, . . .}

• Weight k error events:

Ek = Ek1 =k times︷ ︸︸ ︷E1E1 . . .

• Error algebra:

E = span⋃∞k=1 Ek

Page 26: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Flip Errors

E1 ={I, σx

(1), σx(2), σx

(3), . . .}

• Classical errors + superposition principle.

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Flip Errors

E1 ={I, σx

(1), σx(2), σx

(3), . . .}

• Classical errors + superposition principle.Examples.

0010〉 σx(2)

→ 0110〉

0010〉 σx(1)σx

(3)

→ 1000〉1√2

(0010〉+ 0011〉

)σx

(4)

→ 1√2

(0011〉+ 0010〉

)= 1√

2

(0010〉+ 0011〉

)

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Flip Errors

E1 ={I, σx

(1), σx(2), σx

(3), . . .}

• Classical errors + superposition principle.Examples.

0010〉 σx(2)

→ 0110〉

0010〉 σx(1)σx

(3)

→ 1000〉1√2

(0010〉+ 0011〉

)σx

(4)

→ 1√2

(0011〉+ 0010〉

)= 1√

2

(0010〉+ 0011〉

)• Size of common eigenspaces of σx(i)?

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

Page 30: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→

Page 35: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→ α 10〉+ β 01〉

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→ α 10〉+ β 01〉σx

(2)

−→

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→ α 10〉+ β 01〉σx

(2)

−→ α 01〉+ β 10〉

Page 38: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→ α 10〉+ β 01〉σx

(2)

−→ α 01〉+ β 10〉

σx(1)σx

(2)

−→

Page 39: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Error Detection I

ψ〉ψ〉

L

E

ψ〉L

Successprobability ≤ 1

• Example:

ψ〉 = α 0〉+ β 1〉0〉

L= 00〉

1〉L

= 11〉

00〉〈00 + 11〉〈11

α 00〉+ β 11〉 I(12)

−→ α 00〉+ β 11〉σx

(1)

−→ α 10〉+ β 01〉σx

(2)

−→ α 01〉+ β 10〉

σx(1)σx

(2)

−→ α 11〉+ β 00〉

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Error Detection II

• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉

L, 1〉

L, 2〉

L, . . ..

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Error Detection II

• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉

L, 1〉

L, 2〉

L, . . ..

• C detects E ifPCEPC = λEPC

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Error Detection II

• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉

L, 1〉

L, 2〉

L, . . ..

• C detects E ifPCEPC = λEPC

• Equivalently:

E =

C

C︷ ︸︸ ︷

λE 0 . . . 00 λE

...... . . .0 . . . λE

E12

E21 E22

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Error Detection II

• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉

L, 1〉

L, 2〉

L, . . ..

• C detects E ifPCEPC = λEPC

• Equivalently: For all φ〉L

L〈φ E φ〉L

= λE

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Error Detection II

• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉

L, 1〉

L, 2〉

L, . . ..

• C detects E ifPCEPC = λEPC

• Equivalently: For all φ〉L

L〈φ E φ〉L

= λE

• Equivalently: For all φ〉L, ψ〉

L

φ〉L⊥ ψ〉

L⇒ E φ〉

L⊥ ψ〉

L

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Minimum Distance I

• C has minimum distance ≥ d if

C detects all errors of weight d− 1.

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Minimum Distance I

• C has minimum distance ≥ d if

C detects all errors of weight d− 1.

• C = span(

00〉, 11〉)

is a [[2, 1, 2]]σx(i)

code.

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Minimum Distance I

• C has minimum distance ≥ d if

C detects all errors of weight d− 1.

• C = span(

00〉, 11〉)

is a [[2, 1, 2]]σx(i)

code.

• C is a [[n, k, d]]E1 code means:Length n: Total number of qubits is n.k encoded qubits, dim C = 2k.Minimum distance at least d for E1.

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Minimum Distance II

• Construct a [[3, 1, 3]]σx(i) code — greedily:

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Minimum Distance II

• Construct a [[3, 1, 3]]σx(i) code — greedily:

Add: 0〉L

= 000〉.

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Minimum Distance II

• Construct a [[3, 1, 3]]σx(i) code — greedily:

Add: 0〉L

= 000〉.1〉

Lmust be orthogonal to

000〉100〉, 010〉, 001〉110〉, 101〉, 011〉

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12

Minimum Distance II

• Construct a [[3, 1, 3]]σx(i) code — greedily:

Add: 0〉L

= 000〉.1〉

Lmust be orthogonal to

000〉100〉, 010〉, 001〉110〉, 101〉, 011〉

Choose 1〉L

= 111〉.

Page 52: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

12

Minimum Distance II

• Construct a [[3, 1, 3]]σx(i) code — greedily:

Add: 0〉L

= 000〉.1〉

Lmust be orthogonal to

000〉100〉, 010〉, 001〉110〉, 101〉, 011〉

Choose 1〉L

= 111〉.Encode α 0〉+ β 1〉 → α 000〉+ β 111〉.

• . . . the three bit repetition code.

Page 53: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

13

Error Correction Process

E1 E2 E3

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13

Error Correction Process

E1 E2 E3

• Where is the encoded qubit?

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13

Error Correction Process

E1 E2 E3

• Where is the encoded qubit?Observable algebra always defined:

At = span(I(St), σx

(St), σy(St), σz

(St))

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13

Error Correction Process

E1 E2 E3

• Where is the encoded qubit?Observable algebra always defined:

At = span(I(St), σx

(St), σy(St), σz

(St))

A(S): Algebra between errors and recovery.

A(S) A(S) A(S)

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14

Subsystems

1 2 3 . . .

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14

Subsystems

1 2 3 . . .

• A subsystem is a factor of a subspace of H.

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14

Subsystems

1 2 3 . . .

• A subsystem is a factor of a subspace of H.

• Specifying a subsystem S:• Decompose: H '

(S ⊗ T

)⊕R

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14

Subsystems

1 2 3 . . .

• A subsystem is a factor of a subspace of H.

• Specifying a subsystem S:• Decompose: H '

(S ⊗ T

)⊕R

• Observables: ∗-algebra A(S) ' Matrices(dimS).

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14

Subsystems

1 2 3 . . .

• A subsystem is a factor of a subspace of H.

• Specifying a subsystem S:• Decompose: H '

(S ⊗ T

)⊕R

• Observables: ∗-algebra A(S) ' Matrices(dimS).

• Example with two qubits:Decompose:

0〉S0〉

T= 1√

2

(00〉+ 11〉

), 0〉

S1〉

T= 1√

2

(00〉 − 11〉

)1〉

S0〉

T= 1√

2

(01〉+ 10〉

), 1〉

S1〉

T= 1√

2

(01〉 − 10〉

)

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14

Subsystems

1 2 3 . . .

• A subsystem is a factor of a subspace of H.

• Specifying a subsystem S:• Decompose: H '

(S ⊗ T

)⊕R

• Observables: ∗-algebra A(S) ' Matrices(dimS).

• Example with two qubits:Decompose:

0〉S0〉

T= 1√

2

(00〉+ 11〉

), 0〉

S1〉

T= 1√

2

(00〉 − 11〉

)1〉

S0〉

T= 1√

2

(01〉+ 10〉

), 1〉

S1〉

T= 1√

2

(01〉 − 10〉

)Observables:

{σx

(S) = σx(2)

σz(S) = σz

(1)σz(2)

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Noiseless subsystems

• Subsystem S is noiseless for E if [E ,A(S)] = 0.

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Noiseless subsystems

• Subsystem S is noiseless for E if [E ,A(S)] = 0.

• E = span(I, σx

(1)σx(2), σz

(1))

:

Page 65: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

15

Noiseless subsystems

• Subsystem S is noiseless for E if [E ,A(S)] = 0.

• E = span(I, σx

(1)σx(2), σz

(1))

:

Observables:{σx

(S) = σx(2)

σz(S) = σz

(1)σz(2)

Decompose:

0〉S0〉

T= 1√

2

(00〉+ 11〉

), 0〉

S1〉

T= 1√

2

(00〉 − 11〉

)1〉

S0〉

T= 1√

2

(01〉+ 10〉

), 1〉

S1〉

T= 1√

2

(01〉 − 10〉

)

Page 66: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

15

Noiseless subsystems

• Subsystem S is noiseless for E if [E ,A(S)] = 0.

• E = span(I, σx

(1)σx(2), σz

(1))

:

Observables:{σx

(S) = σx(2)

σz(S) = σz

(1)σz(2)

Decompose:

0〉S0〉

T= 1√

2

(00〉+ 11〉

), 0〉

S1〉

T= 1√

2

(00〉 − 11〉

)1〉

S0〉

T= 1√

2

(01〉+ 10〉

), 1〉

S1〉

T= 1√

2

(01〉 − 10〉

)• E = span

(I, σx

(1)+σx(2)+σx(3), σy(1)+σy(2)+σy(3), σz

(1)+σz(2)+σz(3))

:

Spin 3/2

Spin 1/2

Noiseless qubit (N)

Observables:

{σU

(N) = σ(A) · σ(B)

σV(N) = σ(A) · σ(C)

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Error Correcting Subsystems I

A(S) A(S) A(S)

• Protection requires:

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Error Correcting Subsystems I

A(S) A(S) A(S)

• Protection requires:[ER,A(S)] = 0 R: Any operator of the recovery.

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16

Error Correcting Subsystems I

A(S) A(S) A(S)

• Protection requires:[ER,A(S)] = 0 R: Any operator of the recovery.

• Error correcting subsystem (S, 0〉T):

H '(S ⊗ T

)⊕R, 0〉

T

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Error Correcting Subsystems I

A(S) A(S) A(S)

• Protection requires:[ER,A(S)] = 0 R: Any operator of the recovery.

• Error correcting subsystem (S, 0〉T):

H '(S ⊗ T

)⊕R, 0〉

T

• Usage: 1. Reject R. 2. Reset T to 0〉T. 3. Wait . . .

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16

Error Correcting Subsystems I

A(S) A(S) A(S)

• Protection requires:[ER,A(S)] = 0 R: Any operator of the recovery.

• Error correcting subsystem (S, 0〉T):

H '(S ⊗ T

)⊕R, 0〉

T

• Usage: 1. Reject R. 2. Reset T to 0〉T. 3. Wait . . .

• (S, 0〉T) corrects E if

E ψ〉S

0〉T

= ψ〉SφE〉T

Example.

Page 72: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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Error Correcting Subsystems II

• The three qubit [[3, 1, 3]]σx(i) repetition code.

C = span(

000〉, 111〉)

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17

Error Correcting Subsystems II

• The three qubit [[3, 1, 3]]σx(i) repetition code.

C = span(

000〉, 111〉)

• As a subsystem:0〉

S0〉

T= 000〉 1〉

S0〉

T= 111〉

0〉S

1〉T

= 100〉 1〉S

1〉T

= 011〉0〉

S2〉

T= 010〉 1〉

S2〉

T= 101〉

0〉S

3〉T

= 001〉 1〉S

3〉T

= 110〉

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17

Error Correcting Subsystems II

• The three qubit [[3, 1, 3]]σx(i) repetition code.

C = span(

000〉, 111〉)

• As a subsystem:0〉

S0〉

T= 000〉 1〉

S0〉

T= 111〉

0〉S

1〉T

= 100〉 1〉S

1〉T

= 011〉0〉

S2〉

T= 010〉 1〉

S2〉

T= 101〉

0〉S

3〉T

= 001〉 1〉S

3〉T

= 110〉

• Detection property:

PCσx(i)†σx

(j)PC = δijPC

PC = 0〉TT〈0

Page 75: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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From Correction to Detection

• Suppose (S, 0〉T) corrects errors in E .

Define C by PC = 0〉TT〈0 .

Then C detects every error in E†E .

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From Correction to Detection

• Suppose (S, 0〉T) corrects errors in E .

Define C by PC = 0〉TT〈0 .

Then C detects every error in E†E .

• Proof:S〈ψ T〈0E†D 0〉

Tψ〉

S=

Page 77: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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From Correction to Detection

• Suppose (S, 0〉T) corrects errors in E .

Define C by PC = 0〉TT〈0 .

Then C detects every error in E†E .

• Proof:S〈ψ T〈0E†D 0〉

Tψ〉

S= S〈ψ T〈0E† φ(D)〉

Tψ〉

S

=

Page 78: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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From Correction to Detection

• Suppose (S, 0〉T) corrects errors in E .

Define C by PC = 0〉TT〈0 .

Then C detects every error in E†E .

• Proof:S〈ψ T〈0E†D 0〉

Tψ〉

S= S〈ψ T〈0E† φ(D)〉

Tψ〉

S

= S〈ψ T〈φ(E) φ(D)〉Tψ〉

S

=

Page 79: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

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From Correction to Detection

• Suppose (S, 0〉T) corrects errors in E .

Define C by PC = 0〉TT〈0 .

Then C detects every error in E†E .

• Proof:S〈ψ T〈0E†D 0〉

Tψ〉

S= S〈ψ T〈0E† φ(D)〉

Tψ〉

S

= S〈ψ T〈φ(E) φ(D)〉Tψ〉

S

= S〈ψ ψ〉S

T〈φ(E) φ(D)〉T

= T〈φ(E) φ(D)〉T

= λE†D

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From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

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19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).

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19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).PCE

†iEjPC = λijPC.

Page 83: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).PCE

†iEjPC = λijPC.

Λ = (λij)ij Hermitian⇒ change basis of span(E):λij = δij.

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19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).PCE

†iEjPC = λijPC.

Λ = (λij)ij Hermitian⇒ change basis of span(E):λij = δij.

Let ψ〉L

be a state of C. Defineψ〉

Si〉

T= Ei ψ〉L

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19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).PCE

†iEjPC = λijPC.

Λ = (λij)ij Hermitian⇒ change basis of span(E):λij = δij.

Let ψ〉L

be a state of C. Defineψ〉

Si〉

T= Ei ψ〉L

Choose U unitary: U ψ〉S

0〉T

= ψ〉L.

Page 86: Quantum Error Correction I · 1 LOS ALAMOS National Laboratory One of Two Qubits A ˆ A B State spaces: 0i A + 1i B 00i AB + 01i AB + 10i AB + 11i AB Q ˆ? Q Q

19

From Detection to Correction

• Suppose C detects errors in E†E .There exists (S, 0〉

T) such that for some unitary U :

U†PCU = 0〉TT〈0

(S, 0〉T) corrects span

(EU).

• Proof:span(E) = span(E0, E1, . . . ).PCE

†iEjPC = λijPC.

Λ = (λij)ij Hermitian⇒ change basis of span(E):λij = δij.

Let ψ〉L

be a state of C. Defineψ〉

Si〉

T= Ei ψ〉L

Choose U unitary: U ψ〉S

0〉T

= ψ〉L.

Compute: EiU ψ〉S

0〉T

= Ei ψ〉L = ψ〉Si〉

T

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QEC Theorems

Theorem: Given a quantum system. There exists anerror-correcting subsystem for E iff there exists a E†Edetecting quantum code.

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QEC Theorems

Theorem: Given a quantum system. There exists anerror-correcting subsystem for E iff there exists a E†Edetecting quantum code.

Corollary: Assume E†1 = E1. If C has minimum distance2e+ 1, then C induces an e-error-correcting subsystem.

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QEC Theorems

Theorem: Given a quantum system. There exists anerror-correcting subsystem for E iff there exists a E†Edetecting quantum code.

Corollary: Assume E†1 = E1. If C has minimum distance2e+ 1, then C induces an e-error-correcting subsystem.

• Coding theory lingo: A [[n, k, 2e+ 1]] code is e-errorcorrecting.

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Stabilizer Codes I

• Conventions:X = σx, Y = σy, Z = σz

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Stabilizer Codes I

• Conventions:X = σx, Y = σy, Z = σz

IXIII = σx(2), IIY ZI = σy

(3)σz(4)

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Stabilizer Codes I

• Conventions:X = σx, Y = σy, Z = σz

IXIII = σx(2), IIY ZI = σy

(3)σz(4)

• Pauli group: Products of σu up to sign.

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Stabilizer Codes I

• Conventions:X = σx, Y = σy, Z = σz

IXIII = σx(2), IIY ZI = σy

(3)σz(4)

• Pauli group: Products of σu up to sign.

• Stabilizer of C = span(

000〉, 111〉):

{III, ZZI, IZZ,ZIZ} = 〈 ZZI, IZZ 〉

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Stabilizer Codes I

• Conventions:X = σx, Y = σy, Z = σz

IXIII = σx(2), IIY ZI = σy

(3)σz(4)

• Pauli group: Products of σu up to sign.

• Stabilizer of C = span(

000〉, 111〉):

{III, ZZI, IZZ,ZIZ} = 〈 ZZI, IZZ 〉

• Properties:C detects IXI

IXI · ZZI = −ZZI · IXI

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Stabilizer Codes II

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

• Proof: For ρ ∈ N , let λ(ρ) be the common eigenvalue.

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

• Proof: For ρ ∈ N , let λ(ρ) be the common eigenvalue.σ ∈ N : ok.

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Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

• Proof: For ρ ∈ N , let λ(ρ) be the common eigenvalue.σ ∈ N : ok.σ 6∈ N⊥: Choose ρ ∈ N , σρ = −ρσ.

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22

Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

• Proof: For ρ ∈ N , let λ(ρ) be the common eigenvalue.σ ∈ N : ok.σ 6∈ N⊥: Choose ρ ∈ N , σρ = −ρσ. For ψ〉 in the code:

ρ ψ〉 = λ(ρ) ψ〉

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22

Stabilizer Codes II

• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.

• DefineN⊥ = {Pauli operators which commute with N}.

Theorem: A stabilizer code of N detects all Paulioperators except those in N⊥ \N .

• Proof: For ρ ∈ N , let λ(ρ) be the common eigenvalue.σ ∈ N : ok.σ 6∈ N⊥: Choose ρ ∈ N , σρ = −ρσ. For ψ〉 in the code:

ρ ψ〉 = λ(ρ) ψ〉ρ σ ψ〉 = −σ ρ ψ〉

= −λ(ρ)σ ψ〉 . . .σ ψ〉 is orthogonal to the code.

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The 5 Qubit Code

• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.

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The 5 Qubit Code

• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.

• Example distance check:Every weight ≤ 2 Pauli product UV III is detected:

Y ZI YY IZY

ZY IZZYY ZZI Y Z

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The 5 Qubit Code

• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.

• Example distance check:Every weight ≤ 2 Pauli product UV III is detected:

Y ZI YY IZY

ZY IZZYY ZZI Y Z

Restrict to first two columns:〈 Y Z, IY, Y I, ZY 〉 . . . generates all Pauli products.

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The 5 Qubit Code

• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.

• Example distance check:Every weight ≤ 2 Pauli product UV III is detected:

Y ZI YY IZY

ZY IZZYY ZZI Y Z

Restrict to first two columns:〈 Y Z, IY, Y I, ZY 〉 . . . generates all Pauli products.

⇒ every non-identity UV III anticommutes with a stabilizer.

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The 5 Qubit Code

• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.

• Example distance check:Every weight ≤ 2 Pauli product UV III is detected:

Y ZI YY IZY

ZY IZZYY ZZI Y Z

Restrict to first two columns:〈 Y Z, IY, Y I, ZY 〉 . . . generates all Pauli products.

⇒ every non-identity UV III anticommutes with a stabilizer.

• Rules:X · Y ∼ ZY · Z ∼ XZ ·X ∼ Y

⇔01⊕ 10 = 1110⊕ 11 = 0111⊕ 01 = 10

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