Plethysm is in #BQP

Abstract

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

Figures (4)

• Figure 1: Quantum circuit for the plethysm coefficient $a^{\lambda}_{\mu,\nu}$ (\ref{['thm:plethysm']}). The witness is input in the space $\{\mu\}_{\mathop{\mathrm{GL}}\nolimits(\{\nu\}_{H})}$. The first inverse Schur transform embeds it into $\lvert\mu\rvert$ registers, each isomorphic to $\{\nu\}_{H}$. The second layer of inverse Schur transforms embeds each register into $\lvert\nu\rvert$ registers of dimension $n$. After the final Schur transform, we measure the registers corresponding to the partition and the Weyl module. We accept if and only if the former returns outcome $\lambda$ and the latter $p_0$, where $\ket{p_0}\in\{\lambda\}_{H}$ is an arbitrary fixed basis state of the Weyl module.
• Figure 2: A circuit representation of the algorithm for \ref{['prob:branching_computational']}. The states $\ket{q_0}$ and $\ket{\boldsymbol{\mathbf{q_1}}}$ are arbitrary states used for embedding purposes, and their specific initialization does not impact the algorithm correctness. The permute gate applies the rearrangement described in Step \ref{['step:rearrange']}. For clarity purposes we avoid drawing the registers containing the invariant spaces for the $H$-representations coming from the decomposition of the defining modules for $G$ from Step \ref{['step:measure']} and onward.
• Figure 3: Quantum circuit to establish that a representation-theoretic multiplicity is in $\#\mathsf{BQP}$: Given a witness state in the representation $V$, first embed it into a model representation $V_\text{model}$. Subsequently perform strong Fourier sampling, and accept if the outcomes are $\lambda$ and the label $p$ of an arbitrary fixed basis vector $\ket{p}\in V_\lambda$. If both steps can be done efficiently, the multiplicity is in $\#\mathsf{BQP}$.
• Figure 4: Left: Generalized phase estimation (GPE) circuit (with optional uncomputation omitted). Right: The circuit underlying \ref{['cor:gpe bqp']}. It is a special case of \ref{['fig: embed+SFS']}.

Theorems & Definitions (27)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.2
• proof
• Definition 2.3: Specification of group homomorphism
• Lemma 3.1
• proof
• Definition 4.1: $\mathsf{QMA}$ -- Quantum Merlin-Arthur
• Definition 4.2: $\#\mathsf{BQP}$ -- Counting-$\mathsf{BQP}$