In this short computational essay , we will discuss measurement-based quantum computing as an alternate model to quantum computing . We will show how simple rules of measurements to cluster states can be utilized to generate complex computations. Under certain conditions, we can represent an equivalent circuit for any graph that has subset of vertices(qubits) measured. Finally we will explore briefly with simple examples the measurement calculus, that is a more formal way to do measurement-based quantum computations that also proves is universal.
Special thanks to my colleagues from the Wolfram Quantum Computation Framework Team, Nikolay Murzin and Mads Bahrami for their advice and interesting ideas about this topic.
There are many good references , but the ones i consider very fundamental are :
◼
  • https://arxiv.org/pdf/quant-ph/0503126
    • ◼
  • https://arxiv.org/pdf/0704.1263
    • ◼
  • https://arxiv.org/pdf/quant-ph/0301052

    • Paclet Installation

    Install and load the Wolfram Quantum Computation framework:
    In[]:=
    PacletInstall["https://wolfr.am/DevWQCF",ForceVersionInstall->True]
    Out[]=
    PacletObject
    Name: Wolfram/QuantumFramework
    Version: 1.6.3
    
    In[]:=
    <<Wolfram`QuantumFramework`

    Helper
    Definitions

    Equatorial basis definition :
    In[]:=
    eb=QuantumBasis<|Interpretation["
    θ
    +
    ",0]->{1,Exp[Iθ]}
    2
    ,Interpretation["
    θ
    -
    ",1]->{1,-Exp[Iθ]}
    2
    |>,"Parameters"->θ;
    M operator for doing measurements in CME pattern:
    In[]:=
    m=QuantumMeasurementOperator[Cos[θ]QuantumOperator["X"]+Sin[θ]QuantumOperator["Y"],"Parameter"->θ];​​MM=FullSimplify[m["DiscardExtraQudits"],0<=θ<2Pi];​​M[θ_]:=QuantumOperator[QuantumOperator[Inverse[m[θ]["Eigenbasis"]["Matrix"]]]@MM[θ],"Label"->"Measurement"[m[θ]["Label"]]];

    Graph states and fundamental measurement lemma

    A graph or cluster state is a highly entangled quantum state , which has a direct representation from a mathematical graph
    G
    . The edges
    E
    of the graph represent the entanglement between qubits with a controlled Z operator . The graph state is defined as
    G 〉
    =
    
    0
    G
    〉
    where
    :=
    ∏
    (j,k)ϵE
    C
    j,k
    and
    
    0
    G
    〉
    =
    +〉
    ⊗
    +〉
    ⊗...
    ⊗n
    +〉
    In[]:=
    g=GridGraph[{2,3},ImageSize->Small]
    Out[]=
    The QuantumFramework supports graph states:
    In[]:=
    QuantumState["Graph"[g]]
    Out[]=
    QuantumState
    Pure state
    Qudits: 6
    Type: Vector
    Dimension: 64
    
    Measurement based quantum computing (MBQC) is a universal computing paradigm that makes use of these states, equatorial basis
    (|±α〉=1/
    2
    
    |0 〉
    ±
    α
    
    1〉
    
    ) measurements are really important in MBQC because of a fundamental lemma. Let the state
    | Ψ 〉
    =
    C
    12
    (
    |ψ 〉
    1
    ⊗
    |+ 〉
    2
    )
    with ψ 〉 an arbitrary one qubit state, and apply an equatorial measurement
    M(α)
    to the first qubit. If the outcome is
    m
    then the state of the second qubit is
    
    ψ
    2
    〉
    =
    H
    m
    Z
    RZ
    (α)
    ψ 〉
    =
    m
    X
    H
    RZ
    (α)
    ψ 〉
    Define the ψ state:
    In[]:=
    ψ=QuantumState["RandomPure"];
    Define the Ψ state:
    In[]:=
    Ψ=QuantumOperator["CZ"]@QuantumTensorProduct[ψ,QuantumState["+"]];
    Equatorial state projection:
    In[]:=
    projected=Through[(SuperDagger/@eb[α]["BasisStates"])[Ψ]];
    Transformation of the measurement lemma:
    In[]:=
    transformed=Table[(QuantumOperator["H"]@(QuantumOperator["Z"]^m)@QuantumOperator["RZ"[α]]@ψ),{m,0,1}];
    Verify the equality comparing density matrices,
    2
    factor is included because the projection makes the state not normalized:
    In[]:=
    Chop@FullSimplifyMapThreadNormal[#1["DensityMatrix"]]-Normal[#2["DensityMatrix"]]&,
    2
    projected,transformed,α∈Reals
    Out[]=
    {{{0,0},{0,0}},{{0,0},{0,0}}}
    This lemma can be used in bigger cluster states employing more measurements to implement diverse circuits of any number of qubits. But we can observe that depending of the measurement outcomes Pauli corrections are needed to get the circuit exactly.

    Equivalence of measuring cluster states and implementing circuits

    We will verify symbolically that doing measurements to a graph state has an equivalent circuit representation. In these examples is assumed all the equatorial basis measurements gave as outcome the
    θ
    +
    state (m = 0) , because in that case Pauli corrections are not needed. Usually in experimental photonic MBQC corrections cannot be performed and the desired transformation is post-selected.
    In[]:=
    $Assumptions=Element[{α,β,γ},Reals];

    (3)
    L
    cluster

    Qubits 1 and 2 are measured
    In[]:=
    g=Graph[{1<->2,2<->3},VertexLabels->Automatic]
    Out[]=
    We can use graph states directly using the QuantumFramework:
    In[]:=
    state=QuantumState["Graph"[g]];
    Consider the circuit:
    In[]:=
    q=QuantumCircuitOperator[{"RZ"[α],"RX"[β]}];​​q["Diagram","WireLabels"->{"+〉"}]
    Out[]=
    Verifying the equality:
    In[]:=
    result1=4(QuantumOperator[eb[β]["BasisStates"][[1]]["Dagger"],1]@QuantumOperator[eb[α]["BasisStates"][[1]]["Dagger"],1]@state)["DensityMatrix"]//Normal;
    In[]:=
    result2=(q[QuantumState["Plus"]])["DensityMatrix"]//Normal;
    In[]:=
    FullSimplify[result1-result2]
    Out[]=
    {{0,0},{0,0}}

    (4)
    L
    state

    Qubits 1, 2 and 3 are measured. This configuration can be used to implement an arbitrary single qubit gate
    In[]:=
    g=Graph[{1<->2,2<->3,3<->4},VertexLabels->Automatic]​​state=QuantumState["Graph"[g]];
    Out[]=
    Projectors:
    In[]:=
    projs=eb[#]["BasisStates"][[1]]["Dagger"]&/@{α,β,γ};
    In[]:=
    q=QuantumCircuitOperator[{"RZ"[α],"RX"[β],"RZ"[γ],"H"}];​​q["Diagram"]
    Out[]=
    In[]:=
    result1=8RightComposition[Sequence@@projs][state]["DensityMatrix"]//Normal;
    In[]:=
    result2=ComplexExpand[q[QuantumState["Plus"]]["DensityMatrix"]//Normal];
    In[]:=
    %-%%//Simplify
    Out[]=
    {{0,0},{0,0}}

    (4)
    Horseshoe
    cluster

    Qubits 1 and 2 are measured
    In[]:=
    g=EdgeDelete[GridGraph[{2,2},VertexLabels->Automatic],{4<->3}]​​state=QuantumState["Graph"[g]];
    Out[]=
    In[]:=
    q=QuantumCircuitOperator[{"CZ","RZ"[α],"RZ"[β]->2,"H"->{1,2}}];
    In[]:=
    q["Diagram","WireLabels"->{Style["|+〉",14],Style["|+〉",14]}]
    Out[]=
    In[]:=
    projs=eb[#]["BasisStates"][[1]]["Dagger"]&/@{α,β};
    In[]:=
    result1=4RightComposition[Sequence@@projs][state]["DensityMatrix"]//Normal;
    Qubits 1 and 2 are measured
    Qubits 1 and 2 are measured

    Some extra example

    Qubit 3 will be measured

    Graph to Circuit

    From the previous examples , it is clear that is possible to automate the process of building a circuit from measuring certain qubits of a graph state. To have a direct algorithm it is required the graph follows the methodology of Experimental One-Way Quantum Computing , such that the graph is arranged in a grid-like fashion following the flow of information.

    Examples

    Let’s see some generated circuits using the function:
    Invalid input because the way the graph is arranged would represent a 1 qubit circuit but is not

    CME patterns in MBQC

    There is also a different approach to one-way quantum computing that is based entirely on a semantic calculus of patterns. The calculus incorporates a rewrite system of rules and algebra of pattern composition , patterns can also be extended with tensor product. The most fundamental types of patterns are called CME patterns that are composed of the following commands:

    Linear chain teleportation

    For example in the case of 3 qubits, with computation space {1,2,3} with inputs{1} and outputs {3}, it is possible to show the following pattern implements the teleportation protocol:
    Check the equality:

    CNOT gate

    General one qubit rotation

    A general XZX rotation can also be described with a pattern of computational space {1,2,3,4,5} with inputs {1} and outputs {5} :
    Slight modification to M operator for labeling purposes:
    Verify the pattern:

    CITE THIS NOTEBOOK

    Measurement based quantum computing: equatorial measurement rules, equivalent circuits and patterns​
    by Bruno Tenorio​
    Wolfram Community, STAFF PICKS, May 22, 2024
    ​https://community.wolfram.com/groups/-/m/t/3180183