Namit Anand, Jeffrey Marshall, Jason Saied, Eleanor Rieffel, Andrea Morello
9 min
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.
Alex: So representation theory ties into checking if a set qualifies as a design?
Sam: Yes—think of group actions on spaces breaking into basic unbreakable blocks, called irreducible representations. A set is a t-design if its t-fold copies match the full unitary group's irrep count exactly. For subgroups like Cliffords, this math reveals why they're 2-designs in prime powers but drop to 1-design otherwise.
Alex: And for weighted unitary designs, is there a catch?
Sam: Key result: if a group's weighted version forms a t-design, the unweighted group already does too. This pushes toward universal generators, mixing different sets to hit higher randomness.
Alex: In real quantum hardware like high-spin atoms, what gates do experimenters use to create them?
Sam: High-spin nuclei act like qudits with levels from a spin value S, giving dimension d equals 2S plus one—like a top with more wobble positions. Researchers apply gates called SNAP, which add custom phase twists to each level independently, like tuning the pitch of different strings on a guitar. Pairing SNAP with displacement operations—shifts that rotate the whole spin cloud—lets them build universal control. The paper shows circuits alternating these gates generate approximate t-designs efficiently.
Alex: Spin-coherent states sound native, but do they qualify as designs on their own?
Sam: No, they form only a 1-design, averaging like uniform points on a sphere but failing higher orders. The reason traces to representation theory: the space of two copies decomposes into more independent blocks than a 2-design allows. Continuous rotations share this flaw. That's why circuits mixing SNAP and displacements are needed.
Alex: And for d=6, they give explicit weighted 3-designs from larger Cliffords?
Sam: Yes—for 2-designs, they list states from rigged constructions; for 3-designs, project multiqubit stabilizer states from dimension 8. This works for platforms like superconducting cavities too, using SNAP and echoed displacements.
Alex: Native gates build the uniform starters, projection adapts to any d. How does this feed into benchmarking?
Sam: It enables character randomized benchmarking: decompose the operator space into group irreps—basic symmetry blocks—and fit decay rates per block using a fixed input state. For Cliffords, orbits group by gcd divisors, revealing error types beyond standard RB. The paper suggests this characterizes qudit errors meaningfully where 2-designs were missing.
Alex: Practically, how do they run the experiment with just one starting state?
Sam: They prepare a simple fixed state, like the ground level—like a robot always starting in its home position. Then they run random sequences from the group, sandwiching a test gate in the middle, and measure back in that same basis. Averaging over many runs twirls the errors into blocks that don't mix, letting each symmetry block's decay show separately—like watching paint dry at different steady rates per color.
Alex: Twirling keeps things in their own blocks? And a single state overlaps all of them?
Sam: Yes. The twirl makes it act the same within each block, like stirring soup so lumps stay separate but uniform inside. For single-qudit Cliffords, a fixed state overlaps each via phases, nonzero. So one prep and measure suffice, fitting one decay per divisor of d.
Alex: For any d, even non-prime, this spots error flavors without full 2-designs. What about spin systems?
Sam: Similar for rotations in spin-S qudits. Use projector state at the top spin pole and measure extremes; links ensure nonzero overlaps per block. Fits one decay each.
Alex: And this reveals more than plain RB, since blocks differ?
Sam: Exactly—each irrep decay flags errors symmetric under that block, like distinguishing twist vs. shift flaws. Suggests clearer hardware fingerprints.
Alex: The paper mentions other uses for these weighted designs—like in measurements or learning states?
Sam: Yes, one key application is classical shadow tomography. Figuring out a full quantum state takes tons of measurements, like scanning every pixel of a huge photo. But weighted 2-designs make the measurement set complete enough for qudits, optimal for these shadows; 3-designs help with fidelity checks.
Alex: Does it work for scrambling too, like how info spreads in chaotic systems?
Sam: Right—information scrambling measures how fast local details mix across a system, using out-of-time-ordered correlators. Weighted versions enable qudit scrambling studies where exact groups fall short.
Alex: And for machine learning on quantum circuits?
Sam: In quantum machine learning, training circuits from a 2-design leads to barren plateaus—flat landscapes where changes barely affect the goal. Without 2-designs in non-prime qudits, Cliffords might train better. The paper flags this as a trainability question worth exploring.
Alex: Meaningful extensions, then—shadows, scrambling, learning—all scaled to qudits via weights.
Sam: Yes, the paper proves bounds on circuit depths using native gates like SNAP and displacements, polynomial in t for approximate designs. Overall, a step for qudit protocols.
Alex: Pulling this together, these weighted designs and character methods make benchmarking and tomography possible across any qudit size, without being stuck on prime powers.
Sam: That's the key takeaway. By projecting from larger uniform sets, the paper equips protocols like randomized benchmarking and shadow tomography to work universally, revealing error details through those symmetry blocks.
Alex: Are there practical downsides to these weighted approaches?
Sam: Yes, a notable limitation is size: the projected designs end up larger than the theoretical minimum. Applying the weights requires computing inner products, which measure how much two quantum states overlap—like checking how similar two blurry photos are. These add computational steps, though feasible for current setups.
Alex: Does that limit near-term use?
Sam: It does introduce overhead, but circuit bounds keep generation efficient with native gates. Practically, this means universal calibration across hardware like high-spin nuclei or cavities—a meaningful advance for scaling quantum devices.
Alex: Okay, that balances the trade-offs. It fills a real gap.
Sam: Precisely. The work provides concrete tools for qudit protocols where they were missing, grounded in representation theory and explicit constructions. A solid step forward.
Alex: Thanks, Sam—that's a clear picture of where qudit testing stands now. Thanks for listening to ResearchPod.