Identification of symmetry-protected topological states on noisy quantum computers

Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, topological states that are protected by a symmetry can be identified by distinctive degeneracies in their entanglement spectrum. Here, we provide two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The interchangeability of the two protocols illustrates a deep link between the topological classification of quantum phases of matter and the computational power of their ground states. Both protocols are implemented on an IBM quantum computer and used to identify the topological cluster state. The comparison between the experimental findings and noisy simulations allows us to study the stability of topological states to perturbations and noise.

One of the most important achievements in modern physics is the discovery and classification of topological phases of matter. Topological states do not break any local symmetry and, hence, are robust against local perturbations. In the context of quantum computation, this protection can be used to perform quantum protocols that are robust to local noise sources. The downside of this protection is that local probes are insufficient to identify topological states. Hence, even if one is able to create a topological state, demonstrating its topological character can be very challenging. In this work, we take advantage of the exquisite tunability of superconducting circuits to both realize and identify a family of symmetry-protected topological (SPT) states.
SPT phases can be identified by inspecting their entanglement spectrum (ES), i.e., the set of eigenvalues of the reduced density matrix of a subsystem, ρ A . In particular, for ground states of one dimensional (1D) SPT phases the ES is always formed by degenerate pairs (or multiplets), while in topologically trivial states there is no protected degeneracy [1,2] [64]. This observation stands at the basis of the classification of all SPT phases in one and higher dimensions [3]. A simple explanation for the existence of ES degeneracies is offered by the symmetry-resolved structure of ρ A [4,5]. Consider a SPT phase protected by a unitary symmetry G = G A × G B , where G A and G B act on subsystems A and B, respectively. Because G commutes with the Hamiltonian, the ground state of the SPT phase, ψ gs ⟩, is an eigenstate of the symmetry operator G. When performing a partial trace ρ A = Tr B [ ψ gs ⟩⟨ψ gs ], the conservation of G guarantees that ρ A is block diagonal in G A , see where Π A projects a state on a specific symmetry sector. For simple SPTs, like the Haldane phase of integer spins or Kitaev chains, it was found [6,7] thatρ A that belong to different sectors are identical, leading to a degenerate ES [65].
A related property of SPT phases is the possibility to use their ground states as resources for measurement-based quantum computation (MBQC), where the process of computation is driven by local measurements. This connection was uncovered in Ref. [8] for a measurementbased-adiabatic hybrid. Also, Ref. [9] described a method for realizing 1-qubit unitary logical gates with (non-unit) fidelity above 1/4 in a 1D SPT phase. Ref. [10] established that the quantum-wire-protocol is a uniform property of all ground states belonging to a given SPT phase of 1D spin chains. This result was subsequently extended to include measurementbased quantum gates in 1D SPT phases [11,12] and finally to universal MBQC in 2D SPT phases [13][14][15][16].
Here, we use symmetry-resolved density matrices and MBQC protocols to identify the SPT properties of a quantum state. First, we implement a quantum protocol that accesses each symmetry sector individually. The equivalence of the different sectors helps us identify SPT states and distinguish them from trivial ones. Next, we implement the simplest protocol of quantum information processing in SPT states, namely the quantum wire protocol [10], and experimentally demonstrate its robustness under symmetry-respecting perturbations. The protocol can be disturbed only by perturbations that break the symmetry and make the state trivial, hence providing a complementary method to identify SPT states.

I. CLUSTER STATE
Having in mind the physical realization of our algorithm using qubits, we focus here on the 1D cluster Ising Hamiltonian where {X, Y, Z} are Pauli matrices and h i are referred to as stabilizers [9,[17][18][19][20][21][22][23][24][25][26][27][28]. Its ground state, also known as the 1D cluster state ψ cluster ⟩, is a topological state protected by the Z 2 × Z 2 symmetry associated with the conservation of P odd = ∏ i h 2i+1 = ∏ i X 2i+1 and P even = ∏ i h 2i = ∏ i X 2i . These operators correspond to parities on the sublattices of odd and even sites, respectively. For periodic boundary conditions, the reduced density matrix ρ A (a) ψ cluster ⟩ This transformation can be used to prepare the cluster state in a quantum computer [29,30]: Starting from the 000...⟩ state, one needs to, first, apply Hadamard gates to bring the system to the ground state of H trivial ,

II. SYMMETRY-RESOLVED ENTROPIES
As mentioned in the introduction, we use symmetry-resolved reduced density matrices,ρ A , to identify the SPT nature of the cluster state. A direct measure of these matrices requires an exponentially large number of measurements. We overcome this difficulty by addressing the moments of these matrices,S n = Tr[ρ n A ], which can be measured by realizing n copies of the state [33][34][35][36][37][38][39][40][41][42]. Specifically, for n = 2, this approach is based on the identity (2) The two copies of the cluster states were realized on gray and orange qubits, respectively; The two-qubit gates in red were used to realize SWAP operations between pairs of qubits on the two copies.
Here ρ 2 = ρ⊗ρ is the combined state of two independently prepared copies of a state, and the operator SWAP swaps arbitrary states of the two copies. By applying the SWAP operator only to the subsystem A, one can compute the purity of A, Tr[ρ 2 A ]. Finally, if the SWAP operator is measured along with the projector to the conserved sectors, one can directly obtain the symmetry-resolved entropyS n [37,43,44] [68].
To implement these ideas on a quantum computer, we create two copies of the cluster state with L = 4 qubits using two copies of the circuit of Fig. 2(a). Next, we measure the SWAP operator on each pair of qubits of the two copies, using 4 copies of the quantum circuit introduced by Refs. [31,32], see We first consider the effects of noise on S 2 = Tr[ρ 2 ], see Fig. 3(a). In the presence of noise, the state is not pure and the second Rényi entropy of the full system is ≈ 1.1 × log(2). This value is significantly smaller than the maximally allowed value of 4 × log(2), indicating that the output of the simulation is not trivial. The slope of the entropy changes in the second half of the chain, as in the ideal quantum computer. To study the SPT properties of this noisy state, we compute symmetry-resolved quantities, see Fig. 4. For the trivial state, we find that both the probability and the symmetry-resolved purity are larger for P = +1 than for P = −1. In contrast, in the cluster state the probabilities and purities are identical for the two sectors for all L A < L. Remarkably, the total system (L A = L) is mostly found in the P = +1 state, confirming that the system is targeting the correct pure state.
Using the same QISKIT package, we performed the same calculations on the 15-qubit Mel- ideal, our symmetry resolved probes still correctly identify its SPT nature. One interesting difference between the quantum computer and the noisy simulator can be observed in the symmetry resolved probes of small subsystems, L A = 1, 2. In the actual computer, the two sectors show small, but statistically significant, differences. We identify these errors as due to symmetry-breaking noise sources, such as the aforementioned measurement bias, which were absent in the simulation but present in the physical system. This bias also explains why the Rényi entropy of the L A = 1 subsystem ( Fig. 3(a)) is smaller than 1/2, see Methods section. Our results demonstrate that topological arguments can be used to characterize the main sources of errors and classify them according to their symmetry.

IV. MEASUREMENT-BASED WIRE PROTOCOL
We now turn to the experimental realization of the symmetry-protected wire protocol [10].
In this protocol, a general quantum state is encoded in one boundary of the spin chain.
The state is, then, shuttled to the other boundary in a teleportation-like fashion, by local measurements of the spins along the chain. We apply this protocol to a family of SPT states with Z 2 × Z 2 symmetry, which contains the 1D cluster state as a special case. All states in the family possess the same SPT order and, hence, have the same capacity to transmit one-qubit-worth of quantum information. Our goal is to verify the robustness of the protocol against variation within the phase.
For our implementation on an IBM quantum computer we use the L = 4 cluster state ψ cluster ⟩ described above. The corresponding Z 2 × Z 2 symmetry is generated by P odd = ∏ i=1,3 h i = X 1 X 3 Z 4 and P even = ∏ i=2,4 h i = Z 1 X 2 X 4 , where h i are defined in Eq. 1. The family of SPT states is created applying either symmetry-preserving unitaries U S (α, β) = e iβZ 1 X 2 Z 3 e iαX 3 , or symmetry-breaking unitaries U SB (α, β) = e iβZ 1 X 2 Z 3 e iαY 3 to ψ cluster ⟩. In the former case all resource states respect the Z 2 × Z 2 symmetry and can be continuously connected in a symmetry-respecting fashion to the cluster state. In the latter case, the symmetry is broken and computational uniformity is not guaranteed.
Next, we introduce another qubit realizing the input state ψ in ⟩ and teleport it into the wire by performing a measurement in the 2-qubit cluster basis (a locally rotated Bell basis, { +0⟩ ± −1⟩} ) on ψ in ⟩ and the first qubit of the spin chain, see Fig. 2(d). This particular measurement is chosen to be compatible with the MBQC wire protocol, consisting of local measurements in the X-basis of the remaining qubits, and classically controlled Pauli correction depending on the measurement outcomes. Fig. 5 shows the experimentally measured minimum fidelity f min = min i ⟨ψ i in ρ exp out ψ i in ⟩ for six different input states ψ i in ⟩ and the Pauli-corrected output state ρ exp out resulting from the wire protocol, for the choices β = ±α in both the symmetric and the symmetry-breaking case, see also Appendix VI C. We find that the transmission fidelity is constant as a function of α in the symmetry-respecting case. In the symmetry-breaking case, the transmission fidelity is non-constant as the resource state is varied.

V. CONCLUSION
In this paper we proposed and realized experimentally two algorithms to identify the SPT nature of the cluster state on a quantum computer. The first algorithm stems from the observation that in SPT states, the reduced density matrix ρ A is formed by identical blocks that correspond to different sectors of the underlying symmetry. The flexibility of the quantum computer allowed us to directly probe the moments of density matrices by projecting the quantum state into the different symmetry sectors. The realization of this algorithm on both a quantum simulator and on a IBM quantum computer allowed us to study the impact of time dependent noise on the SPT order of the state. In particular, we found that while most of realistic noise sources are symmetry preserving, the systematic measurement bias of the physical machine breaks this symmetry. Its effects are, however, small enough to enable us to identify the SPT nature of the cluster state. An alternative way to characterize the SPT order of the cluster states consists of using them as a buffer for measurement based quantum teleportation. We find that the fidelity of this protocol is uneffected by symmetry preserving terms, and vice versa for symmetry breaking terms.
Our work has important implications for the modelling of noisy intermediate-scale quantum computers. We have demonstrated that topological arguments are an efficient tool to identify and classify noise sources in quantum computers. This information can be used to improve the performance of quantum computers, for example, by gauging the measurement apparatus to take into account systematic errors. From a fundamental perspective, we identified sufficient conditions under which a noisy quantum state can retain its SPT properties.
This aspect may have implications for quantum computations: for pure states, it was shown that the classification of SPT phases is in one-to-one correspondence with the possibility to use it as a resource for one-way-quantum computer. Although this question deserves further investigation, we conjecture that this link extends to noisy systems as well.
VI. METHODS 1. Quantum algorithm to compute the symmetry resolved purity -The symmetry resolved purity of the subsystem A of size L A < L is defined byS 2 (P ) = Tr[ρ 2 A Π A (P )], where Π A (±1) = (1 ± Y 1 X 2 ...X L A ) 2 is the projection over the P = ±1. We implement this circuit by taking the average between the expectation values of Tr[ρ 2 A ] and Tr[ρ 2 A Y 1 X 2 ...X L A ]. To compute the latter, we implement two copies of the same state, according to Eq. (1). For simplicity, let us focus on a single qubit i, where the operator Tr[ρ 2 A X i ] can be written as Tr[ρ 2 (X i ⊗ I) SWAP i ] and SWAP i swaps the two copies of the qubit i. The operator and eigenvalues {λ i } = {1, −1, i, −i}. This local basis change is performed in Fig. 1(c) using the Z basis and needs to be rotated to the X basis for i > 2 (or the Y basis for i = 1). To obtain Tr[ρ 2 A Π A ], after performing a measurement on each pair of copies and classically recording the appropriate eigenvalue λ i , we perform a quantum average over ∏ L A i=1 λ i . This method generalizes for any moment n and for general symmetry (such as Z N ), hence generalizing the symmetry-resolved entanglement protocols of Refs. [6,37] to qubits.
2. Formal definition of symmetry preserving noise sources -A formal definition of symmetry preserving noise sources can be given by introducing an operator T A , which acts on a subsystem A and maps the different sectors of the symmetry among themselves. In a SPT state, all symmetry-resolved reduced density matrices are identical and hence [T A , ρ A ] = 0. In the example of the cluster state the operators T A flip the edge spins X 1 and X L A and are given by Z 1 and Z L A . A generic noise map Φ ∶ ρ A → ρ ′ A is then said to be symmetrypreserving if it preserves the property [T A , ρ ′ A ] = 0. Specifically, we focus on noise sources that can be described by the Kraus operators according to with the normalization condition ∑ i K † i K i = I, where I is the identity matrix. A trivial example of a symmetry-preserving noise is dephasing, described by the Kraus operators K 1 = √ 1 − pI and K 2 = √ pZ i . Both operators conserve Z i and commute with T A . A nontrivial example is given by the depolarizing noise with K 1 = [(1+ √ 1 − p)I −(1− √ 1 − p)Z i ] 2 and K 2 = √ pσ − i . These operators do not conserve Z and, hence, do not commute with T A . However, because σ − i commutes with the product of two K i , if [ρ, Z i ] = 0 then [ρ ′ , Z i ] = 0 leading to symmetry preservation. These examples highlight the difference between conserved quantities and symmetries: a conserved quantity is always a symmetry, but not vice versa (see Refs. [55][56][57] for an introduction).
3. A simple model of the measurement bias -A natural candidate for the symmetry-breaking noise observed in the quantum computer is a systematic error present in the measurement device, giving preference to state 0 with respect to state , or vice versa. The existence of this error explains why the second Rényi entropy −log[S 2 ] at L A = 1 is smaller than log2, see Fig. 3(a): If we assume that the output qubits are random variables with probabilities 0.5 ± , we obtain −logS 2 = −L A log[(0.5 + ) 2 + (0.5 − ) 2 ] ≈ L A (log2 − 4 2 ). In the same model, the difference between the even and odd probabilities decreases exponentially as S n (P = +1) −S n (P = −1) = (0.5 + ) n − (0.5 − ) n L A ≈ 2 2−n n L A . These expressions are in qualitative agreement with the experimental observations for ≈ 2%, see Figs. 3 and 4.
imental results, are available from the corresponding author upon request.