Random ensembles of symplectic and unitary states are indistinguishable (2409.16500v1)

Abstract: A unitary state $t$-design is an ensemble of pure quantum states whose moments match up to the $t$-th order those of states uniformly sampled from a $d$-dimensional Hilbert space. Typically, unitary state $t$-designs are obtained by evolving some reference pure state with unitaries from an ensemble that forms a design over the unitary group $\mathbb{U}(d)$, as unitary designs induce state designs. However, in this work we study whether Haar random symplectic states -- i.e., states obtained by evolving some reference state with unitaries sampled according to the Haar measure over $\mathbb{SP}(d/2)$ -- form unitary state $t$-designs. Importantly, we recall that random symplectic unitaries fail to be unitary designs for $t>1$, and that, while it is known that symplectic unitaries are universal, this does not imply that their Haar measure leads to a state design. Notably, our main result states that Haar random symplectic states form unitary $t$-designs for all $t$, meaning that their distribution is unconditionally indistinguishable from that of unitary Haar random states, even with tests that use infinite copies of each state. As such, our work showcases the intriguing possibility of creating state $t$-designs using ensembles of unitaries which do not constitute designs over $\mathbb{U}(d)$ themselves, such as ensembles that form $t$-designs over $\mathbb{SP}(d/2)$.

• The paper establishes that Haar random symplectic states form unitary t-designs for all t, disproving previous limits on t > 1.
• The authors employ Weingarten calculus and Brauer algebra representation theory to rigorously verify the statistical equivalence of symplectic and unitary ensembles.
• The findings highlight potential reductions in quantum computational resources for tasks like state tomography and classical shadows through the use of symplectic unitaries.

An Analysis of "Random Ensembles of Symplectic and Unitary States are Indistinguishable"

This paper, authored by Maxwell West, Antonio Anna Mele, Martín Larocca, and M. Cerezo, investigates the indistinguishability of random ensembles of symplectic and unitary states within the field of quantum information. The authors establish that symplectic random states can form unitary state $t$-designs for all $t$. This result contrasts with prior knowledge which asserts that symplectic unitaries fail to be unitary designs for $t > 1$.

Background and Concepts

The paper is predicated upon an intricate understanding of unitary state designs. These are ensembles of pure quantum states that match the moments of a typical state from a $d$-dimensional Hilbert space up to the $t$-th order. Usually, they are derived by evolving a reference pure state using unitaries sampled from the unitary group $\mathbb{U}(d)$. The paper explores whether Haar random symplectic states, evolved using symplectic unitaries from $\mathbb{SP}(d/2)$, can also form unitary state $t$-designs, notwithstanding the known failure of symplectic unitaries to be unitary designs for $t > 1$.

Main Findings

The authors' core finding is that Haar random symplectic states indeed form unitary $t$-designs for all $t$. This implies that their distribution is indistinguishable from that of unitary Haar random states, even when tests use infinite copies of each state. To derive this result, the authors employ the Weingarten calculus and the representation theory of the Brauer algebra.

Their results show that symplectic and unitary random states are statistically indistinguishable, meaning no quantum experiment can discern between a state sampled from a symplectic ensemble and one from a unitary ensemble with a probability greater than $50\%$, even with infinite queries.

Implications and Future Directions

The ability to use symplectic unitaries to generate unitary state designs has profound implications for quantum computing and information. Here are several notable applications and open questions arising from this work:

1. Efficient Approximate State Designs: Generating approximate state designs with symplectic unitaries may require fewer parameters and less computational depth. The paper demonstrates a concrete example where a random circuit using symplectic gates forms an $\epsilon$-approximate 2-design with 60% fewer parameters than its unitary counterpart. This reduction could make quantum operations more practical and efficient.
2. Classical Shadows: The paper identifies that classical shadows, a protocol for estimating properties of unknown quantum states using randomized measurements, can be equivalently implemented using symplectic unitaries. This equivalence opens new pathways for optimizing quantum measurement procedures.
3. State Tomography: The ability to utilize symplectic unitaries for state tomography—reconstructing an unknown quantum state from measurement data—may lead to protocols with optimal sample complexity, up to logarithmic factors. This could further streamline processes in quantum computing where precise state reconstruction is necessary.
4. Symplectic Designs: The recognition that symplectic unitaries can be used to generate unitary state $t$-designs invites exploration into efficiently implementable symplectic $t$-designs. For example, determining if the symplectic Clifford group forms a 3-design over $\mathbb{SP}(d/2)$ could facilitate new and efficient quantum algorithms.
5. Fundamental Insights: Conceptually, the findings challenge the necessity of using unitary group designs for generating state designs. This invites a broader investigation into whether other subgroups within $\mathbb{U}(d)$ can similarly contribute to unitary state designs and what other groups might exhibit this property.

Conclusion

The revealed indistinguishability between random ensembles of symplectic and unitary states is an elegant result that bridges significant gaps in our understanding of quantum state designs. It fosters immediate practical advancements in quantum computing protocols by potentially reducing resource requirements. Furthermore, the underlying mathematical beauty of these results—from the representation theory of the Brauer algebra to the application of the Weingarten calculus—lays a solid foundation for future research. This work not only clarifies theoretical aspects but also paves the way for tangible improvements in quantum technologies.