efficient simulation of quantum computers: the gottesman...
TRANSCRIPT
Efficient simulation of quantum computers: theGottesman-Knill theorem or an application of group
theory to quantum information (part 2)
Vlad Gheorghiu
Department of PhysicsCarnegie Mellon University
Pittsburgh, PA 15213, U.S.A.
January 30, 2008
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 1 / 16
Outline
1 Brief review
2 Stabilizer groupsThe action of the Pauli groupStabilizersConjugation of stabilizer groups under Clifford operations
3 The Gottesman-Knill theorem
4 Simple example
5 References
Both lectures (Wed. Jan 28 and Today, Jan 30) are available onlineat http://quantum.phys.cmu.edu/groupth
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 2 / 16
Brief review
Classical computers
Quantum computers - evolution is unitary, the group U(2n)
|ψ〉final = U|ψ〉initial = U|0, 0, . . . , 0〉
Evolving a quantum state requires in general O(2n) operations!
The Pauli group on one qudit
P1 = {±1,±i}{I ,X ,Y ,Z}
where the Pauli matrices are
X =
(0 11 0
),Y =
(0 −ii 0
),Z =
(1 00 −1
)with XY = iZ and X 2 = Y 2 = Z 2 = I .
The Pauli group on n qudits
Pn = {±1,±i}{X a1Zb1 ⊗ X a2Zb2 ⊗ . . .⊗ X anZbn}
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 3 / 16
Brief review
Classical computers
Quantum computers - evolution is unitary, the group U(2n)
|ψ〉final = U|ψ〉initial = U|0, 0, . . . , 0〉
Evolving a quantum state requires in general O(2n) operations!
The Pauli group on one qudit
P1 = {±1,±i}{I ,X ,Y ,Z}
where the Pauli matrices are
X =
(0 11 0
),Y =
(0 −ii 0
),Z =
(1 00 −1
)with XY = iZ and X 2 = Y 2 = Z 2 = I .
The Pauli group on n qudits
Pn = {±1,±i}{X a1Zb1 ⊗ X a2Zb2 ⊗ . . .⊗ X anZbn}
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 3 / 16
Brief review
Classical computers
Quantum computers - evolution is unitary, the group U(2n)
|ψ〉final = U|ψ〉initial = U|0, 0, . . . , 0〉
Evolving a quantum state requires in general O(2n) operations!
The Pauli group on one qudit
P1 = {±1,±i}{I ,X ,Y ,Z}
where the Pauli matrices are
X =
(0 11 0
),Y =
(0 −ii 0
),Z =
(1 00 −1
)with XY = iZ and X 2 = Y 2 = Z 2 = I .
The Pauli group on n qudits
Pn = {±1,±i}{X a1Zb1 ⊗ X a2Zb2 ⊗ . . .⊗ X anZbn}
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 3 / 16
Brief review
Classical computers
Quantum computers - evolution is unitary, the group U(2n)
|ψ〉final = U|ψ〉initial = U|0, 0, . . . , 0〉
Evolving a quantum state requires in general O(2n) operations!
The Pauli group on one qudit
P1 = {±1,±i}{I ,X ,Y ,Z}
where the Pauli matrices are
X =
(0 11 0
),Y =
(0 −ii 0
),Z =
(1 00 −1
)with XY = iZ and X 2 = Y 2 = Z 2 = I .
The Pauli group on n qudits
Pn = {±1,±i}{X a1Zb1 ⊗ X a2Zb2 ⊗ . . .⊗ X anZbn}
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 3 / 16
Brief review
The Clifford group C1 on one qubit
C1P1C†1 = P1
The Clifford group Cn on n qubits
CnPnC†n = Pn
Up to complex phases, the Clifford group Cn is generated by H,S andCNOT , where
H =1√2
(1 11 −1
),S =
(1 00 i
)and CNOT =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
For example, HXH† = Z , CNOT (X ⊗ I )CNOT † = X ⊗ X .
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 4 / 16
Brief review
The Clifford group C1 on one qubit
C1P1C†1 = P1
The Clifford group Cn on n qubits
CnPnC†n = Pn
Up to complex phases, the Clifford group Cn is generated by H,S andCNOT , where
H =1√2
(1 11 −1
),S =
(1 00 i
)and CNOT =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
For example, HXH† = Z , CNOT (X ⊗ I )CNOT † = X ⊗ X .
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 4 / 16
Brief review
The Clifford group C1 on one qubit
C1P1C†1 = P1
The Clifford group Cn on n qubits
CnPnC†n = Pn
Up to complex phases, the Clifford group Cn is generated by H,S andCNOT , where
H =1√2
(1 11 −1
),S =
(1 00 i
)and CNOT =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
For example, HXH† = Z , CNOT (X ⊗ I )CNOT † = X ⊗ X .
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 4 / 16
Brief review
The Clifford group C1 on one qubit
C1P1C†1 = P1
The Clifford group Cn on n qubits
CnPnC†n = Pn
Up to complex phases, the Clifford group Cn is generated by H,S andCNOT , where
H =1√2
(1 11 −1
),S =
(1 00 i
)and CNOT =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
For example, HXH† = Z , CNOT (X ⊗ I )CNOT † = X ⊗ X .
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 4 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms
1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;
2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.
We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)
2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)
3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)
4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.
Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
The action of a group G on a set A is defined as a binary function
G × A −→ A
denoted by (g , a) −→ g · a that satisfies the following two axioms1 (gh) · a = g · (h · a) for all g , h ∈ G and a ∈ A;2 e · a = a for every a ∈ A (here e is the identity of G ).
Now let G be the Pauli group P1 and A the Hilbert space of 1 qudit,C2.
A basis of C2 is formally denoted by {|0〉, |1〉}.We define an action of the Pauli group on the Hilbert space of onequbit by specifying how the elements of G act on the basis
1 I |0〉 = |0〉, I |1〉 = |1〉 (identity)2 X |0〉 = |1〉, X |1〉 = |0〉 (bit flip)3 Z |0〉 = |0〉, Z |1〉 = −1|1〉 (phase flip)4 Y |0〉 =?, Y |1〉 =?.
Now we know how G is acting on any element of C2, by linearity, e.g.X (α|0〉+ β|1〉) = α|1〉+ β|0〉.Check that we have a valid action function.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 5 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}
Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)?
The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups The action of the Pauli group
Similarly, we define the action of the Pauli group on n qudits Pn onthe Hilbert space of n qudits Cn
2
First note that a basis of Cn2 is formally specified as
{|0, 0, . . . , 0〉, |0, 0, . . . , 1〉, . . . , |1, 1, . . . , 1〉}Recall that an element g of Pn consists of a phase times a tensorproduct of n Pauli matrices
Each component of g acts individually on the correspondingcomponent of the basis element
Example: consider 3 qubits, let g = X ⊗ I ⊗ Z be an element of P3
and let us see how it acts on |1, 0, 1〉; thenX ⊗ I ⊗ Z |1, 0, 1〉 = −|0, 0, 1〉. Extend the action by linearity.
So now we know how Pn acts on the Hilbert space Cn2 of n qubits
What is (Z ⊗ X ⊗ X )(α|0, 0, 1〉+ β|1, 1, 0〉)? The result isα|0, 1, 0〉 − β|1, 0, 1〉.
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 6 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
We now have an action of Pn on the Hilbert space Cn2
Suppose S is a subgroup of Pn
Define VS to be the set of n qubit states which are fixed by everyelement of S , i.e.
VS = {|ψ〉 ∈ Cn2 : s|ψ〉 = (+1)|ψ〉,∀s ∈ S}
VS is the vector space stabilized by S , and S is called the stabilizer ofVS
Convince yourself that VS is a vector space (show that an arbitrarylinear combination of elements from VS also belongs to VS )
Show that VS is the intersection of the subspaces fixed by eachoperator in S (that is, the eigenvalues one eigenspaces of elements ofS)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 7 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?
Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.
Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn?
Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.
The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}
S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
Example: what is the vector space stabilized by {I ,X} ⊂ P1?Answer: {|0〉}.Another example: what about {I ⊗ I ,Z ⊗ Z} ⊂ Pn? Answer:Span{|0, 0〉, |1, 1〉}.The last one... {I ,Z1Z2,Z2Z3,Z1Z3} stabilizesSpan{|0, 0, 0〉, |1, 1, 1〉}S and VS are dual to each other, in the sense that one uniquelydetermines the other and vice versa.
Clever idea: one can describe (some) subspaces (VS ) by subgroups(S). What’s the advantage?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 8 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian
2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
The necessary and sufficient conditions that S must satisfy in order tostabilize a nontrivial vector space VS are
1 S must be Abelian2 −I does not belong to S
The necessity is easy...
Describe the stabilizer group S of some subspace VS by specifying itsgenerators
A group of size K has at most log(K ) generators!
So a stabilizer subspace of n qubits can be specified by a set of kindependent generators, and it turns out that k is always smaller thann
In the last example from the previous slide, S was generated by〈Z1Z2,Z2Z3〉 = 〈Z ⊗ Z ⊗ I , I ⊗ Z ⊗ Z 〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 9 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S?
Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.
So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ...
... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ...
... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ...
.......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... ..........
〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Stabilizers
Theorem
Let S = 〈g1, g2, . . . , gk〉 be a stabilizer group generated by k independentand commuting generators from Pn, such that −I /∈ S. Then the vectorspace stabilized by S has dimension 2n−k .
Enlarging S reduces the dimension of VS
What if k = n?
Stabilizer states
A stabilizer group generated by n independent generators stabilizes aone-dimensional vector space. The latter is called a stabilizer state.
Example: Let S = 〈Z ⊗ I , I ⊗ Z 〉. So n = 2, k = 2. What is the statestabilized by S? Answer: |00〉.So the state |0, 0, . . . , 0〉 is a stabilizer state with stabilizer groupgenerated by ... ... ... .......... 〈Z1,Z2, . . . ,Zn〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 10 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =c |ψ〉.If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =c |ψ〉.If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =
c |ψ〉.If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =c |ψ〉.
If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =c |ψ〉.If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
Stabilizer groups Conjugation of stabilizer groups under Clifford operations
Remember that the Clifford group maps Pauli operators to Paulioperators
Let c ∈ Cn. Note that S ′ = cSc† is also a stabilizer group, and thedimension of VS ′ equals the dimension of VS . Why?
Convince yourself that if S stabilizes a stabilizer state |ψ〉, thenS ′ = cSc† stabilizes |ψ′〉 =c |ψ〉.If we know how c conjugates the generators of S , we then know howc conjugates the whole S . Why?
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 11 / 16
The Gottesman-Knill theorem
We can now state and prove the following theorem
The Gottesman-Knill theorem
A quantum unitary evolution that uses only the following elements can besimulated efficiently on a classical computer:
1 preparation of qubits in computational basis states (w.l.o.g. can be taken tobe |0, 0, . . . , 0〉 )
2 evolution U from the Clifford group
3 measurements in the computational basis.
The proof is now quite simple...
The first assumption guarantees that we start with a stabilizer state|ψ〉, with stabilizer group Sψ generated by n independent generators〈Z1,Z2, . . . ,Zn〉Now let’s pick a generator of the Clifford group, call it u
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 12 / 16
The Gottesman-Knill theorem
We can now state and prove the following theorem
The Gottesman-Knill theorem
A quantum unitary evolution that uses only the following elements can besimulated efficiently on a classical computer:
1 preparation of qubits in computational basis states (w.l.o.g. can be taken tobe |0, 0, . . . , 0〉 )
2 evolution U from the Clifford group
3 measurements in the computational basis.
The proof is now quite simple...
The first assumption guarantees that we start with a stabilizer state|ψ〉, with stabilizer group Sψ generated by n independent generators〈Z1,Z2, . . . ,Zn〉Now let’s pick a generator of the Clifford group, call it u
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 12 / 16
The Gottesman-Knill theorem
We can now state and prove the following theorem
The Gottesman-Knill theorem
A quantum unitary evolution that uses only the following elements can besimulated efficiently on a classical computer:
1 preparation of qubits in computational basis states (w.l.o.g. can be taken tobe |0, 0, . . . , 0〉 )
2 evolution U from the Clifford group
3 measurements in the computational basis.
The proof is now quite simple...
The first assumption guarantees that we start with a stabilizer state|ψ〉, with stabilizer group Sψ generated by n independent generators〈Z1,Z2, . . . ,Zn〉Now let’s pick a generator of the Clifford group, call it u
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 12 / 16
The Gottesman-Knill theorem
We can now state and prove the following theorem
The Gottesman-Knill theorem
A quantum unitary evolution that uses only the following elements can besimulated efficiently on a classical computer:
1 preparation of qubits in computational basis states (w.l.o.g. can be taken tobe |0, 0, . . . , 0〉 )
2 evolution U from the Clifford group
3 measurements in the computational basis.
The proof is now quite simple...
The first assumption guarantees that we start with a stabilizer state|ψ〉, with stabilizer group Sψ generated by n independent generators〈Z1,Z2, . . . ,Zn〉
Now let’s pick a generator of the Clifford group, call it u
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 12 / 16
The Gottesman-Knill theorem
We can now state and prove the following theorem
The Gottesman-Knill theorem
A quantum unitary evolution that uses only the following elements can besimulated efficiently on a classical computer:
1 preparation of qubits in computational basis states (w.l.o.g. can be taken tobe |0, 0, . . . , 0〉 )
2 evolution U from the Clifford group
3 measurements in the computational basis.
The proof is now quite simple...
The first assumption guarantees that we start with a stabilizer state|ψ〉, with stabilizer group Sψ generated by n independent generators〈Z1,Z2, . . . ,Zn〉Now let’s pick a generator of the Clifford group, call it u
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 12 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉
Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Under the action of u, the initial stabilizer state |ψ〉 stabilized bysome Sψ will evolve to u|ψ〉Remember the duality between a stabilizer state and a stabilizer group
Now |ψ′〉 = u|ψ〉 can be as well described by its stabilizer groupSψ′ = uSψu†
To specify Sψ′ , it is enough to see how the n generators of Stransform under conjugation by u, so we need to finduZ1u
†, . . . , uZnu†
To summarize, implementing a generator u of the Clifford groupimplies updating the n generators that describe the initial quantumstate
This step requires O(n2) operations on a classical computer
In conclusion, if our unitary evolution U is a product of m terms, eacha generator of the Clifford group, then the computation can besimulated by a classical computer in time O(mn2)
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 13 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
The Gottesman-Knill theorem
Intuitively, the way the classical computer performs the simulation issimply to keep track of the generators of the stabilizer as the variousClifford elementary operations are being performed in the computation
At the end, we are left with some stabilizer group that stabilizes somestate
Knowing the stabilizer is equivalent to knowing the state (rememberthe duality)
End of story...
It is clear that one cannot perform universal quantum computationwith only Clifford operations and measurements in the computationalbasis
It is not at all obvious that Clifford operations together with only onequbit unitary that does not belong to the Clifford group form a denseset in U(2n), that is, one can implement universal quantumcomputing using this gates!
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 14 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution
1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉
After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉
After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉
The final stabilizer defines |ψfinal〉 = |00〉+|11〉√2
, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
Simple example
Simulate the evolution of the state |ψinitial〉 = |00〉 throughU = CNOT12H1
The following table is all we need for simulating any Clifford evolution1 Conjugation by CNOT : X1 → X1X2,X2 → X2,Z1 → Z1,Z2 → Z1Z2
2 Conjugation by H: X → Z ,Z → X3 Conjugation by S : X → Y ,Z → Z4 Conjugation by X : X → X ,Z → −Z5 Conjugation by Y : X → −X ,Z → −Z6 Conjugation by Z : X → −X ,Z → Z
|00〉 ⇔ 〈Z ⊗ I , I ⊗ Z 〉After H, 〈X ⊗ I , I ⊗ Z 〉After CNOT , 〈X ⊗ X ,Z ⊗ Z 〉The final stabilizer defines |ψfinal〉 = |00〉+|11〉√
2, so we simulated the
evolution without computing U|ψinitial〉
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 15 / 16
References
1 Michael A. Nielsen and Isaac L. Chuang, Quantum Computationand Quantum Information, Cambridge University Press (2000)
2 Daniel Gottesman, PhD Thesis, arXiv:quant-ph/9705052, preprint
Vlad Gheorghiu (CMU) Efficient simulation of quantum computers January 30, 2008 16 / 16