Post on 29-Sep-2020
Quantum Error Correction IProtecting Quantum Information
Manny
• Qubits are subsystems.
• Error control methods.
• Algebraic error models.
• Error detection.
• Error correction.
• Stabilizer codes.
1LOS ALAMOSNational Laboratory
One of Two Qubits
A ⊂ A B
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
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), . . .
2
Passive Error Control
A B
• Example noise operators:I, σx
(B), σy(B), σz
(B).
2
Passive Error Control
A B
• Example noise operators:I, σx
(B), σy(B), σz
(B).
Information in A is protected.
2
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
2
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).
3
Active Error Control
A B
• Example noise operators:I, cxnot = σx
(B)cnot(BA).
3
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
3
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.
3
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
3
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
4
Error Control Methods
• Passive error control.Noiseless subsystems.
4
Error Control Methods
• Passive error control.Noiseless subsystems.
• Error suppression.Refocusing.Active symmetryenforcement.Decoupling.Reservoir engineering.
4
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.
4
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.
5
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]
6
The Pauli Error Model
1 2 3 . . .• Error operators:
E1 ={I, σx
(1), σy(1), σz
(1), σx(2), σy
(2), . . .}
6
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), . . .}
6
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
6
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.
7
Algebraic Error Models
• Weight 1 error events:E1 = { I, E1, E2, . . .}
7
Algebraic Error Models
• Weight 1 error events:E1 = { I, E1, E2, . . .}
• Weight k error events:
Ek = Ek1 =k times︷ ︸︸ ︷E1E1 . . .
7
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
8
Flip Errors
E1 ={I, σx
(1), σx(2), σx
(3), . . .}
• Classical errors + superposition principle.
8
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〉
)
8
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)?
9
Error Detection I
ψ〉ψ〉
L
E
ψ〉L
Successprobability ≤ 1
9
Error Detection I
ψ〉ψ〉
L
E
ψ〉L
Successprobability ≤ 1
• Example:
ψ〉 = α 0〉+ β 1〉0〉
L= 00〉
1〉L
= 11〉
9
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
9
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)
−→
9
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〉
9
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)
−→
9
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〉
9
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)
−→
9
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〉
9
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)
−→
9
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〉
10
Error Detection II
• A quantum code is a subspace C.Projection operator: PC.Logical basis: 0〉
L, 1〉
L, 2〉
L, . . ..
10
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
10
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
10
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
10
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
11
Minimum Distance I
• C has minimum distance ≥ d if
C detects all errors of weight d− 1.
11
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.
11
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.
12
Minimum Distance II
• Construct a [[3, 1, 3]]σx(i) code — greedily:
12
Minimum Distance II
• Construct a [[3, 1, 3]]σx(i) code — greedily:
Add: 0〉L
= 000〉.
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〉
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〉.
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.
13
Error Correction Process
E1 E2 E3
13
Error Correction Process
E1 E2 E3
• Where is the encoded qubit?
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))
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)
14
Subsystems
1 2 3 . . .
14
Subsystems
1 2 3 . . .
• A subsystem is a factor of a subspace of H.
14
Subsystems
1 2 3 . . .
• A subsystem is a factor of a subspace of H.
• Specifying a subsystem S:• Decompose: H '
(S ⊗ T
)⊕R
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).
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〉
)
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)
15
Noiseless subsystems
• Subsystem S is noiseless for E if [E ,A(S)] = 0.
15
Noiseless subsystems
• Subsystem S is noiseless for E if [E ,A(S)] = 0.
• E = span(I, σx
(1)σx(2), σz
(1))
:
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〉
)
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)
16
Error Correcting Subsystems I
A(S) A(S) A(S)
• Protection requires:
16
Error Correcting Subsystems I
A(S) A(S) A(S)
• Protection requires:[ER,A(S)] = 0 R: Any operator of the recovery.
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
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 . . .
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.
17
Error Correcting Subsystems II
• The three qubit [[3, 1, 3]]σx(i) repetition code.
C = span(
000〉, 111〉)
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〉
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
18
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 .
18
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=
18
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
=
18
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
=
18
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
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).
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, . . . ).
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.
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.
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
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.
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
20
QEC Theorems
Theorem: Given a quantum system. There exists anerror-correcting subsystem for E iff there exists a E†Edetecting quantum code.
20
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.
20
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.
21
Stabilizer Codes I
• Conventions:X = σx, Y = σy, Z = σz
21
Stabilizer Codes I
• Conventions:X = σx, Y = σy, Z = σz
IXIII = σx(2), IIY ZI = σy
(3)σz(4)
21
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.
21
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 〉
21
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
22
Stabilizer Codes II
22
Stabilizer Codes II
• A stabilizer code (Pauli code, symplectic code) is acommon eigenspace of a commutative subgroup Nof the Pauli group.
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}.
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 .
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.
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.
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.
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〉
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.
23
The 5 Qubit Code
• Minimum distance 3 code for Pauli errors:Stabilizer: 〈 Y ZZY I, IY ZZY, Y IY ZZ,ZY IY Z 〉.
23
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
23
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.
23
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.
23
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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