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arXiv:2211.05127 · Physical Review X

Continuous-Variable Quantum State Designs: Theory and Applications

Joseph T. Iosue, Kunal Sharma, Michael J. Gullans, Victor V. Albert

Published Feb 8, 2024• 14 citations
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Abstract

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t -designs from [Blume-Kohout , ], no state designs exist for t≥2 . Similarly, we prove that no CV unitary t -designs exist for t≥2 . We propose an alternative definition for CV state designs, which we call rigged t -designs, and provide explicit constructions for t=2 . As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024

DOI
10.1103/physrevx.14.011013

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We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t -designs from [Blume-Kohout , ], no state designs exist for t≥2 . Similarly, we prove that no CV unitary t -designs exist for t≥2 . We propose an alternative definition for CV state designs, which we call rigged t -designs, and provide explicit constructions for t=2 . As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024
We investigate state designs for continuous-variable quantum systems using the aid of latticelike quantum states. These are code states of Gottesman-Kitaev-Preskill (GKP) codes. We show that the set of all GKP states forms a rigged continuous-variable state 2-design for an n-mode system. We use these lattice state designs to construct a continuous variable shadow tomography protocol, derive sample complexity bounds for both global and local GKP shadows under reasonable physical assumptions, and provide the physical gadgets needed to implement this protocol.
We introduce an ε\varepsilonε-approximate unitary 2-design that is compatible with the structure of p- and q-quadratures in continuous-variable (CV) quantum systems. The design unitaries are defined on a finite-dimensional discretisation of the CV space and can be physically implemented as operations on the full CV space. This establishes the first approximate unitary design for CV systems. The design alternatingly acts with unitaries based on the quadrature operators q^\hat qq^​ and p^\hat pp^​. We prove that the parameter ε\varepsilonε is given by 1/dℓ1/d^\ell1/dℓ, where ddd is the dimension of the truncated Hilbert space and ℓ\ellℓ is the number of iterations. We propose an Unclonable Encryption scheme in which the encryption operators are given by the unitaries which constitute the approximate unitary design. We prove its security using recent results on decoupling. This establishes unclonable-indistinguishable security for a CV encryption for the first time.
In this article we develop a continuous variable (CV) shadow tomography scheme with wide ranging applications in quantum optics. Our work is motivated by the increasing experimental and technological relevance of CV systems in quantum information, quantum communication, quantum sensing, quantum simulations, quantum computing and error correction. We introduce two experimentally realisable schemes for obtaining classical shadows of CV (possibly non-Gaussian) quantum states using only randomised Gaussian unitaries and easily implementable Gaussian measurements such as homodyne and heterodyne detection. For both schemes, we show thatN=O(poly (1 /ϵ, log (1/δ),Mr+αn, log(m) )) samples of an unknownm-mode state ρ suffice to learn the expected value of anyr-local polynomial in the canonical observables of degree α, both with high probability 1 - δ and accuracy ϵ, as long as the state ρ has moments of ordern> α bounded byMn. By simultaneously truncating states and operators in energy and phase space, we are able to overcome new mathematical challenges that arise due to the infinite-dimensionality of CV systems. We also provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes, by leveraging recent energy-constrained entropic continuity bounds. Finally, we provide numerical evidence of the efficiency of our protocols in the case of CV states of relevance in quantum information theory, including ground states of quadratic Hamiltonians of many-body systems and cat qubit states. We expect our scheme to provide good recovery in learning relevant states of 2D materials and photonic crystals.
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.
Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to nmaxn_{\mathrm{max}}nmax​ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as O(nmax4)\mathcal{O}(n_{\mathrm{max}}^4)O(nmax4​), improving upon previous O(nmax13/3)\mathcal{O}(n_{\mathrm{max}}^{13/3})O(nmax13/3​) bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.
Toric t -designs, or equivalently t -designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t -designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense Btmodm sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state 2 -designs in dimension d of size exactly d(d+1) that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension 6 (and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.