Matchgate circuit representation of fermionic Gaussian states: optimal preparation, approximation, and classical simulation

TL;DR

This work derives lower bounds on the number of gates required to prepare an arbitrary pure FGS and establishes both an asymptotic bound on the minimal gate count over general nearest-neighbor gate sets and an exact bound for circuits composed solely of matchgates.

Abstract

Fermionic Gaussian states (FGSs) and the associated matchgate circuits play a central role in quantum information theory and condensed matter physics. Despite being possibly highly entangled, they can still be efficiently simulated on classical computers. We address the question of how to optimally create such states when using matchgate circuits acting on product states. To this end, we derive lower bounds on the number of gates required to prepare an arbitrary pure FGS: We establish both an asymptotic bound on the minimal gate count over general nearest-neighbor gate sets and an exact bound for circuits composed solely of matchgates. We present explicit algorithms whose constructions saturate these bounds, thereby proving their optimality. We furthermore determine when an FGS can be prepared with a circuit of any given depth, and derive an algorithm that constructs such a circuit whenever this condition is satisfied, either exactly or approximately. Our results have direct applications to (approximate) state preparation and to disentangling procedures. Moreover, we introduce a new classical simulation algorithm for matchgate circuits, based entirely on manipulating the generating circuits of the FGSs. Finally, we briefly study an extension of our framework for $t$-doped Gaussian states and circuits.

Figures (6)

• Figure 1: Results and algorithms. Panels (a) and (b) (lower panel) summarize the algorithms to determine matchgate circuits (MGCs) that generate a given fermionic Gaussian state (FGS): (a) In the symmetric Euler decomposition, matchgates are selected to eliminate entries in the covariance matrix (CM). Pairs of qubits are successively removed and the circuit produced is in right standard form (RSF). (b) FGSs whose CM is $d$-banded can be prepared with MGCs of depth $\mathcal{O}(d)$. For determining low-depth circuit for states with (approximately) banded CM, the entanglement cutting algorithm produces small MGCs that cut the state into two subsystems. Panels (c), (d) and (e) show applications of RSF circuits and the algebraic matchgate identities (GYB and LR relation in the top right panel): (c) Since any FGS admits a representation as an RSF circuit, comparing RSF circuits for the same FGS allows one to identify the circuit with minimal gate count. (d) Efficient evaluation of the inner product of two FGS (including phase information) by rewriting the inner product as a simple tensor network contraction. (e) Standard forms similar to the RSF can be derived for circuits, states and inner product evaluations, where the initial circuit is an MCG interleaved with $t$ resourceful gates (resourceful gates are depicted in red, whereas matchgates are blue).
• Figure 2: (a) Action of the symmetric Euler decomposition algorithm for $q=1$ on the covariance matrix $\Gamma$ of a pure FGS on four qubits. Blank fields indicate zeros whereas grey fields mark nonzero entries. The applied Givens rotations correspond to the matchgates shown above the arrows. (b) The procedure to combine two diagonals of gates outputted by the symmetric Euler decomposition algorithm into a single diagonal of matchgates.
• Figure 3: (a) Absorption algorithm: The GYB and LR relations can be used to absorb gates into a matchgate circuit MoLa25. In this example, the circuit is and remains in the maximal RSF. (b) Inner product algorithm: We start with a matchgate circuit in RSF. In step (i), the routine invert_diagonal, which sequentially applies the LR move from left to right (see main text), is applied to the highlighted gates, effectively flipping the corresponding structure. For step (ii), each but the last gate is absorbed into the resulting RSF circuit on two fewer qubits using the routine to_rsf. Steps (iii) and (iv) are repetitions of the first two steps for smaller circuits. The inner product finally is evaluated using tensor network techniques.
• Figure 4: Standard forms for $t$-doped (a) matchgate circuits, (b) states, and (c) inner products. The standard form for the circuits are obtained by applying the GYB relation to general $t$-doped matchgate circuits (i.e., matchgate circuits interleaved with $t$ resourceful non-matchgates). For states, the depth is further reduced by the application of the LR move in a similar way as for the absoption algorithm. Finally, the inner product circuit is found by also using the inverted LR move with the other product state.
• Figure 5: (a) Illustration of Observation \ref{['observation:threepartite_bore']}. In case party $\mathbf{B}$ comprises sufficiently many qubits, any FGS with $\beta$-banded CM is equivalent to some entangled pairs shared between parties $\mathbf{A}$ and $\mathbf{B}$, and $\mathbf{B}$ and $\mathbf{C}$ up to matchgate circuits acting on the parties. (b) Illustration of the algorithm to determine the circuit $U_\mathbf{B}$. The matchgate circuit $U_\text{diag}$ is chosen to diagonalize the reduced CM on party $\mathbf{B}$. The second circuit, $U_\text{perm}$, permutes the entangled qubits such each qubit is adjacent to the party to which it is entangled

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• proof
• Definition 1: Non-commuting product notation
• Lemma 1: Generalized triangle Euler decomposition (see e.g. Refs. JiSu18KiMc18DaDe19OsDa22RaRu69HoRa72)
• proof
• Lemma 2: Brickwall decomposition of MGC OsDa22CaKo22
• Lemma 3
• proof
• proof : Proof of Observation \ref{['obs:symplectic_yb']}