Entropic lens on stabilizer states

The $n$-qubit stabilizer states are those left invariant by a $2^n$-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically motivated generating set, the Hadamard, phase, and controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the “reachability graphs,” at $n\le 5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by cnot gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and cnot gates acting on only two of the $n$ qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and nonholographic states within the stabilizer reachability graphs.

Figure 1

The collection of subgraph structures up through five qubits. (a) One-qubit reachability diagram displaying all single-qubit operations. (b) Two-qubit restricted graph consisting of subgraph structures $g_{24}$ and $g_{36}$. (c) Two new subgraph structures occur at three qubits $g_{144}$ and $g_{288}$. (d) At four qubits we witness the arrival of a new subgraph $g_{1152}$, as well as an increase of entropy vectors on $g_{144}$. No new subgraph structures emerge beyond $g_{1152}$, but the number of different entropy vectors present in each subgraphs continues to increase for higher-qubit number. At four qubits we witness the first instance of nonholographic states, appearing on subgraph $g_{144}$.

Figure 2

Complete one-qubit reachability diagram. Because the Hadamard gate is its own inverse we have depicted edges corresponding to Hadamard gate applications as undirected, but gates corresponding to phase gates as directed.

Figure 3

The one-qubit reachability diagram restricted to only phase (top) and only Hadamard (bottom) operations reveals disconnected orbits of varying length for states $|\mathrm{\Psi }\rangle \in S_1$.

Figure 4

This complete two-qubit reachability graph depicts the full map between all stabilizer states under action of the Clifford group. Edges depicting $H_i$ and ${\mathrm{CNOT}}_{i,j}$ are undirected since they are each their own inverse. $P$ is not its own inverse since $P^4=I$; consequently, the phase gates are represented by directed edges. The color of the edge indicates the type of gate, and the line texture (solid vs dashed) indicates the qubits which the gate acts on.

Figure 5

The 14 squares and 2 connected pair subgraphs in the graph restricted to $H_1$ and $H_2$ at two qubits. Since $[H_1,H_2]=0$ and $H_1^2=H_2^2=\mathbb{1}$, these subgraphs are the only allowed shapes. The four states which connect in pairs rather than squares are given in Eq. (15).

Figure 6

Removing the set of cnot operations from the full reachability graph Fig. 4 reveals two disconnected subgraphs. Since only the cnot gates can change the entropy, each subgraph has the same entropy vector for all of its states.

Figure 7

Subgraph of two-qubit complete reachability graph restricted the subgroup generated by cnot and phase operations. Included structures are various attachments of the simpler structures found in the only-phase and only-cnot restricted graphs in Figs. 19 and 20 in Appendix pp1.

Figure 8

The two-qubit subgraph restricted to $H_1,H_2,{\mathrm{CNOT}}_{1,2},$ and ${\mathrm{CNOT}}_{2,1}$ has two subgraphs which are connected only via phase gates. Trivial loops have been removed in this representation of the two-qubit $H_1,H_2,{\mathrm{CNOT}}_{1,2},{\mathrm{CNOT}}_{2,1}$ restricted graph.

Figure 9

Hamiltonian path $\mathcal{C}$ beginning on state $|i,1\rangle$ (encircled in green) and ending on state $|-,-i\rangle$ (encircled in red).

Figure 10

Pictured are the two unique subgraph types that occur in the three-qubit restricted graph of only $H_1,H_2,$ and $H_3$ operations. Degenerate pairs in the two-qubit $H_1,H_2$ restricted graph (Fig. 5) are promoted to boxes at three qubits. Boxes from the two-qubit $H_1,H_2$ graph are promoted to cubes by the addition of the $H_3$ gate.

Figure 11

This figure depicts the four unique subgraphs which arise in the full three-qubit $H_1,H_2,{\mathrm{CNOT}}_{1,2},{\mathrm{CNOT}}_{2,1}$ restricted graph, pictured in Fig. 23 of Appendix pp1. Each subgraph is labeled by its corresponding vertex count. The full graph consists of six copies of $g_{24}$ and $g_{36}$, three copies of $g_{144}$, and a single copy of $g_{288}$.

Figure 12

Lift of Hamiltonian path $\mathcal{C}$ defined in Eq. (24), from the two-qubit $g_{36}$ subgraph to a copy of $g_{144}$ at three qubits. The path begins on $|\psi \rangle \equiv {\mathrm{CNOT}}_{1,3}|i10\rangle$, the state lifted in Eq. (29).

Figure 13

Four copies of Hamiltonian path $\mathcal{C}$ that give a vertex cover of $g_{144}^2$. Each path is indicated by the lifted starting state where it begins.

Figure 14

The Hamiltonian cycle on $g_{144}$ can be mapped to a cycle on $g_{288}$ through a cnot operation connecting the two subgraphs. Two copies of $g_{144}$ can be used to cover $g_{288}$ completely.

Figure 15

At four qubits, the $g_{144}$ subgraph can have four different entropy vectors among the vertices. The cyan vertices denote states with entropy vector $\vec{S}=(1,1,1,1,1,1,1)$, an entropic structure that violates monogamy of mutual information (14). These four-qubit stabilizer states, located on four-qubit subgraphs $g_{144}$ and $g_{288}$, are the first instances of nonholographic states.

Figure 16

At four qubits, we witness a variation to the $g_{288}$ subgraph first seen at three qubits. The structure is isomorphic to its three-qubit counterpart; however, the number of entropy vectors present has increased to four, similar to what occurred in Fig. 15.

Figure 17

One of the two $g_{1152}$ occurring at four qubits. This subgraph is isomorphic to the $g_{1152}$ subgraph in Fig. 18, but with only two different entropy vectors among all the states in the subgraph.

Figure 18

Another copy of the $g_{1152}$ subgraph introduced at four qubits. States located on this subgraph exhibit four different entropy vectors, in contrast to the $g_{1152}$ subgraph in Fig. 17.

Figure 19

The two-qubit $P_1,P_2$ restricted graph contains five unique substructures. The four isolated points are states on which both $P_1$ and $P_2$ act trivially. The boxlike structures come in two varieties. The box of unentangled states has a trivial loop at each corner, while the boxes of entangled states witness degenerate action instead. There exist two largest structures of states (top left) on which both phase gates act nontrivially and nondegenerately.

Figure 20

The subgroup generated by ${\mathrm{CNOT}}_{1,2}$ and ${\mathrm{CNOT}}_{2,1}$ partitions the set of two-qubit states into three graph substructures. The isolated points are states on which both cnot gates act trivially. The linear triplets are built of one state, on which both cnot gates act nontrivially, connected to two states on which opposing one cnot gate acts trivially. These triplets can occur with states of similar or different entropic structure. The largest structure is a hexagon which illustrates the maximum cycle of the subgroup generated by ${\mathrm{CNOT}}_{1,2}$ and ${\mathrm{CNOT}}_{2,1}$.

Figure 21

Complete reachability diagram on three qubits with trivial loops removed. The Hadamard and phase gates act individually on all three qubits, while the cnot gate acts on any pair of qubits. Line texture indicates the particular action, e.g., a solid line for $H_1$ and medium dashed line for $H_2$.

Figure 22

There are 11 unique subgraphs whose copies build the three-qubit $P_1,P_2,P_3$ restricted graph. These subgraphs segregate according to which sequences of $P_1,P_2$, and $P_3$ are equivalent.

Figure 23

The graph contains six copies of $g_{24}$ and $g_{36}$, three copies of $g_{144}$, and a single copy of $g_{288}$. The three $g_{144}$ subgraphs are isomorphic, but were generated with slightly different layouts by the software used to build this graph.