Graph state measurement

src/graphs/cphase.jl

@@ -45,6 +45,7 @@ Perform local complementation on pure graph
 """
 function local_comp!(g, id::Int)
     # inneighbors or outneighbors shouldn't really matter because the graph is undirected
+    # NOTE: Here we don't need to copy nghbr even though it returns a reference because local complementation guarantees not changing it
     nghbr = inneighbors(g, id)
     for i1 in eachindex(nghbr)
         for i2 in i1+1:length(nghbr)

src/graphs/graphs.jl

@@ -5,7 +5,7 @@ export graphstate, graphstate!, graph_gatesequence, graph_gate
 import Graphs: Graphs, nv, AbstractGraph
 using QuantumClifford: AbstractStabilizer, AbstractSingleQubitOperator,
         sId1, sInvPhase, sSQRTX, CliffordOperator, SingleQubitOperator,
-        Stabilizer, sZ, sPhase, sX, sY, sHadamard, sInvSQRTX, @S_str, stabilizerview,
+        Stabilizer, sZ, sPhase, sX, sY, sHadamard, sInvSQRTX, sInvSQRTY, sSQRTY, @S_str, stabilizerview,
         canonicalize_gott!, phases, sCPHASE, affectedqubits, canonicalize!, tab
 import QuantumClifford: nqubits, apply!
 
@@ -282,5 +282,6 @@ end
 
 include("./two_qubits_table.jl")
 include("./cphase.jl")
+include("./project.jl")
 
-end #module
+end #module

src/graphs/project.jl

@@ -0,0 +1,156 @@
+# Some documentation for myself:
+# In line with the existing definition
+# - `projectrand!(gs::GraphState, m::sMX)` doesn't care about the register/classical bit
+# - `apply!(state::Register, m::sMX)` does, and it depends on `projectrand!`
+# And these should be the only public facing APIs for now
+
+function remove_neighbors!(g, qubit::Int)
+    for nghbr in copy(inneighbors(g, qubit))
+        rem_edge!(g, qubit, nghbr)
+    end
+end
+
+"""
+Project a pure graph state (all VOPs are identity) onto eigenspaces of the Z observable.
+"""
+function _projectZ_pure_graph!(state::GraphState, qubit::Int, res::Bool)
+    # For a = qubit to measure, b ∈ nghbrs(a)
+    # Cₐ -> Cₐ * Xₐ^res * Hₐ
+    # Cᵦ -> Cᵦ * Zᵦ^res
+    # And remove all edeges between a and its neighbors
+    g = graph(state)
+    v = vops(state)
+
+    # Tweak VOPs
+    if res
+        _apply_vop_right!(v, sX(qubit))
+    end
+    _apply_vop_right!(v, sHadamard(qubit))
+    for nghbr in inneighbors(g, qubit)
+        if res
+            _apply_vop_right!(v, sZ(nghbr))
+        end
+    end
+
+    # Modify graph
+    remove_neighbors!(g, qubit)
+
+    return state
+end
+
+"""
+Project a pure graph state onto eigenspaces of the Y observable.
+"""
+function _projectY_pure_graph!(state::GraphState, qubit::Int, res::Bool)
+    g = graph(state)
+    v = vops(state)
+
+    function tweak_vop(q)
+        if res
+            _apply_vop_right!(v, sInvPhase(q))
+        else
+            _apply_vop_right!(v, sPhase(q))
+        end
+    end
+
+    # Tweak VOPs of target qubit and its neighbors
+    tweak_vop(qubit)
+    for b in inneighbors(g, qubit)
+        tweak_vop(b)
+    end
+
+    # perform local complementation on edge
+    local_comp!(g, qubit)
+
+    # remove the neighbors of the target qubit, this step is missing in the paper
+    remove_neighbors!(g, qubit)
+
+    return state
+end
+
+"""
+Project a pure graph state onto eigenspaces of X observable.
+"""
+function _projectX_pure_graph!(state::GraphState, qubit::Int, res::Bool)
+    g = graph(state)
+    v = vops(state)
+    a = qubit
+
+    # Cₐ -> Cₐ Z^res
+    if res
+        _apply_vop_right!(v, sZ(a))
+    end
+
+    nghbrs_a = copy(inneighbors(g, a))
+    if !isempty(nghbrs_a)
+        b = nghbrs_a[1]
+        nghbrs_b = copy(inneighbors(g, b))
+        nghbr_a_and_b = intersect(nghbrs_a, nghbrs_b)
+
+        # tweak the VOPs first
+        # C_b -> C_b (sInvSQRTY)(†) where † is applied only if res is true
+        if res
+            _apply_vop_right!(v, sSQRTY(b))
+        else
+            _apply_vop_right!(v, sInvSQRTY(b))
+        end
+
+        if res
+            # nghbr(b) \ nghbr(a) \ {a}
+            for c in setdiff(nghbrs_b, nghbrs_a, a)
+                _apply_vop_right!(v, sZ(c))
+            end
+        else
+            # nghbr(a) \ nghbr(b) \ {b}
+            for c in setdiff(nghbrs_a, nghbrs_b, b)
+                _apply_vop_right!(v, sZ(c))
+            end
+        end
+
+        # complementation on edges {{c, d} | c ∈ nghbr(a), d ∈ nghbr(b)}
+        for (c, d) in Iterators.product(nghbrs_a, nghbrs_b)
+            if (c ∈ nghbrs_b) && (d ∈ nghbrs_a)
+                if c < d
+                    toggle_edge!(g, c, d)
+                end
+            else
+                toggle_edge!(g, c, d)
+            end
+        end
+
+        # complementation on edges {{c, d} | c, d ∈ nghbr(a) ∩ nghbr(b)}
+        for c in eachindex(nghbr_a_and_b)
+            for d in c+1:length(nghbr_a_and_b)
+                toggle_edge!(g, nghbr_a_and_b[c], nghbr_a_and_b[d])
+            end
+        end
+
+        # complement on edges {{b, d} | d ∈ nghbr(a) \ {b}}
+        for d in nghbrs_a[2:end]
+            toggle_edge!(g, b, d)
+        end
+
+        remove_neighbors!(g, a)
+    end
+    return state
+end
+
+function _project_graph!(state::GraphState, qubit::Int, op::Stabilizer, res::Bool)
+    # Cₐ† opₐ Cₐ
+    op_prime = apply!(op, inv(vops(state)[qubit]))
+
+    ispositive = (phases(op_prime)[1]) != 0x02
+    # normalize the phase to match later
+    phases(op_prime)[1] = 0x00
+    res_tilde = ispositive ? res : !res
+
+    if op_prime == S"Z"
+        _projectZ_pure_graph!(state, qubit, res_tilde)
+    elseif op_prime == S"X"
+        _projectX_pure_graph!(state, qubit, res_tilde)
+    elseif op_prime == S"Y"
+        _projectY_pure_graph!(state, qubit, res_tilde)
+    end
+
+    return state
+end

test/test_graphs_project.jl

@@ -0,0 +1,24 @@
+@testitem "Graph states single-qubit Pauli measurements" begin
+    using Random
+    import QuantumClifford.GraphSim: _project_graph!, GraphState
+
+    test_sizes = [2,10,63,64,65,127,128,129]
+    iteration = 1000
+
+    @testset "X measurement" begin
+        for n in test_sizes
+            s = random_stabilizer(n)
+            for _ in 1:iteration
+                g = GraphState(s)
+                q = rand(1:n)
+                # println("state: ", s)
+                # println("Graph state: ", Stabilizer(g))
+                # println("qubit: ", q)
+                _, res = projectXrand!(s, q)
+                # println("res: ", res == 0x02)
+                _project_graph!(g, q, S"X", res == 0x02)
+                @test canonicalize!(Stabilizer(g)) == canonicalize!(copy(s))
+            end
+        end
+    end
+end