Preparing matrix product states via fusion: constraints and extensions

Abstract

In the era of noisy, intermediate-scale quantum (NISQ) devices, the efficient preparation of many-body resource states is a task of paramount importance. In this paper we focus on the deterministic preparation of matrix-product states (MPS) in constant depth by utilizing measurements and classical communication to fuse smaller states into larger ones. We place strong constraints on the MPS that can be prepared using this method, which we refer to as MPS fusion. Namely, we establish that it is necessary for the MPS to have a flat entanglement spectrum. Using the recently introduced split-index MPS (SIMPS) representation, we then introduce a family of states that belong to interesting phases of matter protected by non-onsite symmetries, including anomalous and non-invertible symmetries, and also serve as resources for long-range quantum teleportation, but which lie beyond the scope of ordinary MPS fusion. It is shown constructively that these states can be prepared in constant depth using a broader class of measurement-assisted protocols, which we dub SIMPS fusion. Even in cases when MPS fusion is possible, using SIMPS fusion can give rise to significantly reduced resource overhead. We also discuss constraints on SIMPS fusion and propose a general framework for fusion that encompasses the MPS and SIMPS protocols. Our results therefore simultaneously establish the boundaries of conventional MPS fusion and push the envelope of which states can be prepared using measurement-assisted protocols.

Figures (5)

• Figure 1: Fusing two copies of the AKLT state. Undesirable measurement outcomes result in the red Pauli operators, which are corrected using the symmetries \ref{['eqn:AKLT-symm']} of the MPS tensor. (a) Initial state: two short, disconnected AKLT chains. (b) Fuse the two intervening boundary legs using a Bell measurement. Depending on the measurement outcome, a Pauli operator $P$ is inserted on the fused leg, shown for the case $P=X$. This erroneous operator can be pushed to the boundary using onsite $U_P$ operators [Eq. \ref{['eqn:AKLT-symm']}]. Finally, these operators can be removed by applying the corresponding $U_X$ and boundary $X$ operators.
• Figure 2: Demonstrating the global symmetry $\bar{U}_{\mathsf{CZ}}$\ref{['eqn:UCZ-OBC']} of the normal SIMPS with OBC. First, we use the trick in Eq. \ref{['eqn:delta_trick']} to pull the $\mathsf{CZ}$ operators down. Then, we use the second symmetry in Eq. \ref{['eqn:nice-simps-symmetries']} to push these onto the virtual legs, where the $X$'s cancel pairwise.
• Figure 3: The protocol for fusing two copies of the normal SIMPS. We want to fuse two short SIMPS into a single longer one. The protocol has two steps. In the first step, we fuse the upper legs using a $\mathsf{CX}$ gate and a $Z$-basis measurement. When the measurement outcome is $s=1$, an $X$ operator is inserted into the tensor network as shown. The $X$ operator can be turned into a $\mathsf{CZ}$ using Eq. \ref{['eqn:nice-simps-symmetries']}, which can then be removed by using the trick in Eq. \ref{['eqn:delta_trick']} and acting with $\mathsf{CZ}$ on the appropriate legs. In the second step, we fuse the lower legs using a Bell measurement. Depending on the measurement outcome, a Pauli operator $P$ is inserted on the fused leg. For the case when $P=X$, this operator can be pushed to the boundary degree of freedom by applying the global symmetry $U_{\mathsf{CZ}}$ as in Fig. \ref{['fig:nice_simps_global_symm']}. Undoing the $\mathsf{CZ}$ operators and the boundary $X$ completes the fusion. The case for $P=Z,Y$ are similar, except $P$ is pushed to the boundary using $Z$ operators or $Z$ and $\mathsf{CZ}$ operators, respectively. Note that the first step can also be implemented unitarily as discussed in the main text.
• Figure 4: Generalized fusion of MPS which encompasses both MPS fusion and SIMPS fusion. Two states of size $L$ with boundary conditions defined by $\mathbb{P}$ and $\mathbb{Q}$ are fused to a state of size $2L+C$ with the same boundary conditions.
• Figure 5: The generalized push-through relations considered in Appendix \ref{['app:asymmetric-pushthrough']}, pictured for $N=4$. The unitary operator $U$ ($\tilde{U}$) is used to push $V$ to the left (right) boundary of the MPS.