Qudit Designs and Where to Find Them

TL;DR

This work introduces a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions, and establishes bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits.

Abstract

Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.

Figures (6)

• Figure 1: State Welch test ratio $R_w$ (ratio of left to right hand side in Eq. \ref{['eq:welch-test-weighted-design']}) minus one, where $R_w-1=0$ for a state $t$-design. The four different panels show different qudit dimension $d=6,9$, and the approach to a $t=2,3$ design, with the number of states $N$ used. The legend gives the circuit depth, where each layer is a random real displacement $D(\theta, \phi=0)$ ($\theta\in [0,\pi]$ selected according to density $\frac{1}{2}\sin \theta$), followed by a random SNAP gate (angles selected uniformly in $[0, 2\pi)$). The initial state is $|S,S\rangle$. For deep enough circuits, the curves fall like $R_w-1\sim 1/N$.
• Figure 2: Numerical evaluation of the relative error between the Haar integral $I(d,t):=\int_{\mathcal{U}(d)} |\mathrm{tr}(U)|^{2t} d\mu_U$ and the Gamma function $\Gamma(t+1)$, for $d=3,4,6$. This is computed by taking the trace of $10^8$ Haar random unitary samples (using scipy), and numerically estimating the integral. The random looking form of these curves is likely explained by finite sampling effects, however, the broad features are fairly transparent: We see in the regime $t \le d$, the approximation is fairly accurate (relative error less than 1% in all cases). Also notice the error generally reduces increasing $d$, consistent with the expectation that in the asymptotic limit, $I$ does indeed converge to $\Gamma(t+1)$. We observe the error does not appear to go to zero with the number of samples, suggesting, as per the main text, that the integral value is not exactly $\Gamma(t+1)$, even for $t\le d$.
• Figure 3: Fractional frame potential ratio for one and two qubit Clifford group.
• Figure 4: Frame potential ratio for a qubit (two-dimensional) representation of $SL(2,\mathbb{F}_5)$, which is a 5-design.
• Figure 5: Fractional frame potential ratio for cyclic groups $C_d$ of order $d$ (see legend), with respect to $U(d)$. One can compute $R_f(d,t) = d^{2t-1}/\Gamma(t+1)$. For $d\rightarrow \infty$, the convergence point $t^*\rightarrow 1/2$ (see main text).

Theorems & Definitions (25)

• Definition 1: Equivalent ways of defining unitary $t$-designs
• Lemma 1: Bounds on frame potential
• Theorem 1: Iterating tensor product expanders generates unitary designs nakata_efficient_2017
• Definition 2: Welch test for weighted state designs
• Lemma 2: Bounds on the cardinality of weighted unitary designs
• Lemma 3: Equivalence of state designs, SIC-POVMs, and MUBs klappenecker_mutually_2005
• Proposition 1: gross_evenly_2007zhu_clifford_2016mele_introduction_2023
• Definition 3
• Theorem 2
• proof