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Quick Quantum Computing with Q3
Q3 provides many utilities and mechanism to facilitate the study of quantum computation and information. In this document, we explain some basic usages of Q3 in the context of quantum computation and quantum information are explained.
A species | |
Ket | Represents a quantum state of a system of species |
Basis | Generates the basis of the Hilbert space associated with a system of species |
Matrix | Gives the matrix representation of operators on the Hilbert space associated with a system of species |
ExpressionFor | Recovers an operator expression from a matrix representation. |
Elaborate | Transforms an expression to a more explicit form. |
Represents a measurement of Pauli operators (including tensor products of single-qubit Pauli operators) | |
Analyses the probabilities and post-measurement states of a Pauli measurement on a quantum state. |
Q3 functions useful in quantum computation and quantum information.
Make sure that the Q3 package is loaded to use the demonstrations in this document.
In[90]:=
Needs["QuantumMob`Q3mini`"]
Throughout this document, symbol S will be used to denote qubits and Pauli operators on them. Similarly, symbol c will be used to denote complex-valued coefficients.
In[1]:=
Let[Qubit,S]Let[Complex,c]
Quantum States in Q3
A quantum bit, or qubit for short, is a quantum two-level system. It is the elementary unit of quantum computers. Each qubit is associated with a two-dimensional Hilbert space (≃ ). The computational basis of the Hilbert space is denoted by .
2
{|0〉,|1〉}
The Hilbert space associated with a system of qubits is constructed with the tensor product of the Hilbert spaces of individual qubits and is isomorphic to ⊗⊗⋯⊗=. The standard tensor-product basis of the Hilbert space is given by
n
2
2
2
⊗n
()
2
{|〉⊗|〉⊗⋯⊗|〉:=0,1}
s
1
s
2
s
n
s
k
Any quantum state in the Hilbert space can be expanded in this basis as
|ψ〉
|ψ〉==0=0⋯=0|〉⊗|〉⊗⋯⊗|〉
1
∑
s
1
1
∑
s
2
1
∑
s
n
s
1
s
2
s
n
c
s
1
s
2
⋯s
n
for some complex coefficients .
c
s
1
s
2
⋯s
n
Q3 provides two different ways to represent the computational basis states of an -qubit system.
n
◼
Using to represent tensor-product state . Although it looks simple, this method becomes quickly tedious as the number of qubits increases. This method is suitable for a small system of qubits.
|〉⊗|〉⊗⋯⊗|〉
s
1
s
2
s
n
◼
Using represent tensor-product state . It may look complicated at the first glance but is more efficient for a larger system of qubits.
|〉⊗|〉⊗⋯⊗|〉
s
1
s
2
s
n
Method Using Ket[…]
Method Using
Ket[…]
For example, consider a three-qubit system. Here is the standard tensor-product basis of the system.
In[5]:=
bs=Basis[3]
Out[6]=
{|0,0,0〉,|0,0,1〉,|0,1,0〉,|0,1,1〉,|1,0,0〉,|1,0,1〉,|1,1,0〉,|1,1,1〉}
In[7]:=
InputForm[bs]
Out[7]//InputForm=
{Ket[{0, 0, 0}], Ket[{0, 0, 1}], Ket[{0, 1, 0}], Ket[{0, 1, 1}], Ket[{1, 0, 0}],
Ket[{1, 0, 1}], Ket[{1, 1, 0}], Ket[{1, 1, 1}]}
Ket[{1, 0, 1}], Ket[{1, 1, 0}], Ket[{1, 1, 1}]}
A state vector is a linear superposition of the standard basis states.
In[8]:=
ket=Ket[{0,1,0}]-I*Ket[{1,0,0}]
Out[8]=
|0,1,0〉-|1,0,0〉
A state vector can be represented by a column vector in the standard tensor-product basis.
In[9]:=
vec=Matrix[ket];vec//MatrixForm
Out[10]//MatrixForm=
0 |
0 |
1 |
0 |
- |
0 |
0 |
0 |
In[11]:=
new=ExpressionFor[vec]
Out[11]=
|0,1,0〉-|1,0,0〉
In[12]:=
new-ket
Out[12]=
0
Method Using
Method Using
In this method, you need to first choose a symbol, say, S, to denote a group of qubits.
In[13]:=
Let[Qubit,S]
Symbol S declared as qubit can take flavor indices in the form . The preceding flavor indices ,,… specifies a particular qubit in the class denoted by the symbol S. The last flavor index refers to the Pauli operator (, respectively, to be discussed below) on the qubit specified by the preceding indices ,,…. An exceptional case is , and refers to the qubit itself (rather than an operator) specified by the indices ,,….
S[,,…,μ]
j
1
j
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1
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μ=0,1,2,3,…
I,X,Y,Z,…
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μ=
$
S[,,…,]
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1
j
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$
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Now, a computational basis state is specified by
For example, the logical basis for qubit S[1,$] is comprised of these two states.
In[14]:=
v0=Ket[S[1,$]->0]
v1=Ket[S[1,$]->1]
v1=Ket[S[1,$]->1]
Out[14]=
0
S
1
Out[15]=
1
S
1
In[16]:=
{v0,v1}//InputForm
Out[16]//InputForm=
{Ket[<|S[1, $] -> 0|>], Ket[<|S[1, $] -> 1|>]}
For a two-qubit system, here is the standard tensor-product basis. Recall that different qubits (within the same class referred to by symbol S) are distinguished by the flavor indices.
In[17]:=
bs=Basis[S[1],S[2]]bs//InputForm
Out[17]=
,,,
0
S
1
0
S
2
0
S
1
1
S
2
1
S
1
0
S
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1
S
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2
Out[18]//InputForm=
{Ket[<|S[1, $] -> 0, S[2, $] -> 0|>], Ket[<|S[1, $] -> 0, S[2, $] -> 1|>],
Ket[<|S[1, $] -> 1, S[2, $] -> 0|>], Ket[<|S[1, $] -> 1, S[2, $] -> 1|>]}
Ket[<|S[1, $] -> 1, S[2, $] -> 0|>], Ket[<|S[1, $] -> 1, S[2, $] -> 1|>]}
Typing in the special flavor index $ every time you specify a state vector would be cumbersome, and within , it can be dropped. (Many other functions in Q3 also allow it.)
In[19]:=
v0=Ket[S[1]->0]
v1=Ket[S[1]->1]
v1=Ket[S[1]->1]
Out[19]=
0
S
1
Out[20]=
1
S
1
In[21]:=
ket=v0-Iv1KetRegulate[ket,{S[1,$],S[2,$]}]KetRegulate[ket,{S[1],S[2]}]
Out[21]=
-
0
S
1
1
S
1
Out[22]=
-
0
S
1
0
S
2
1
S
1
0
S
2
Out[23]=
-
0
S
1
0
S
2
1
S
1
0
S
2
A state vector can be represented by a column vector in the standard tensor-product basis (computational basis). This can be achieved by . For example, consider the following state.
Matrix
In[24]:=
ket=S[1,6]**S[2,6]**Ket[]
Out[24]=
1
2
0
S
1
0
S
2
1
2
0
S
1
1
S
2
1
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S
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2
Get the column-vector representation of it. Here note that is equivalent to inside .
S[{1,2}]={S[1],S[2]}
S[{1,2},]={S[1,],S[2,]}
$
$
$
Matrix
In[25]:=
vec=Matrix[ket,S@{1,2}];vec//MatrixForm
Out[26]//MatrixForm=
1 2 |
1 2 |
1 2 |
1 2 |
Operators on Qubits in Q3
Like state vectors, Q3 provides two different ways to represent the Pauli operators.
Consider a tensor product of single-qubit Pauli operators.
Consider a state vector.
You can act operator op on the above state vector using Multiply or, equivalently, **.
The operator expression may be recovered with ExpressionFor.
Consider a state vector.
You can act operator op on the above state vector using Multiply (**).
It is a good habit to check the standard basis implied in the matrix representation.
Use the above sets to construct some composite expressions. For example, you can construct
the Heisenberg Hamiltonian for two spins (qubits).
the Heisenberg Hamiltonian for two spins (qubits).
Measurements in Q3
Q3 supports measurement of any Pauli operators (including tensor products of single-qubit Pauli operators).
As a simple example, consider a three-qubit system in the following state.
Performing the measurement, you can get random outcomes according to the probabilities shown above.
Collecting the random outcomes, you can infer the probability distribution.
As in the above example, one can infer the probability distribution by sampling the outcomes.
Quantum Circuit Model
Some frequently used quantum circuit elements in the quantum circuit model of quantum computation.
Here is an simple example of quantum circuit.
One can convert a quantum circuit to an analytic expression.
One can also directly apply a quantum circuit to an input state.
The above output state should be the same as the result from acting operator expression op.
It is possible to multiply the quantum circuit with an operator as well.