[Paper Review] Continuous-variable quantum state designs: theory and applications

Joseph T. Iosue, Kunal Sharma+2 more|arXiv (Cornell University)|Nov 9, 2022

Quantum Computing Algorithms and Architecture126 references4 citations

TL;DR

This paper introduces rigged continuous-variable (CV) quantum state designs—non-normalizable states in a rigged Hilbert space—to overcome the non-existence of standard CV state $t$-designs for $t \geq 2$. It constructs explicit $t=2$ rigged designs using Fock and phase states, enabling a design-based shadow tomography protocol and linking torus $2$-designs to complete sets of mutually unbiased bases (MUBs).

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 Comm. Math. Phys. 326, 755 (2014), no state designs exist for $t\geq2$. Similarly, we prove that no CV unitary $t$-designs exist for $t\geq 2$. 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.

Motivation & Objective

• To address the fundamental limitation that standard continuous-variable (CV) quantum state $t$-designs do not exist for $t \geq 2$ in infinite-dimensional Hilbert spaces.
• To develop a meaningful generalization of quantum designs for CV systems by relaxing the normalizability condition.
• To enable new applications in quantum information, such as shadow tomography and fidelity estimation, in continuous-variable quantum systems.
• To establish a connection between torus $2$-designs and complete sets of mutually unbiased bases (MUBs) in finite-dimensional Hilbert spaces.
• To define an energy-constrained average fidelity for CV quantum channels and relate it to the CV entanglement fidelity.

Proposed method

• Introduces rigged $t$-designs using non-normalizable states in the dual space $S(\mathbb{R})'$, extending the Hilbert space to include tempered distributions.
• Constructs explicit $t=2$ rigged designs using Fock states and phase states, which form a positive operator-valued measure (POVM) for phase estimation.
• Develops a design-based shadow tomography protocol for CV states using rigged $2$-designs to estimate expectation values of low-degree polynomial observables.
• Defines energy-constrained rigged designs to enable average fidelity estimation for CV quantum channels.
• Uses the connection between simplex designs and state designs to prove the non-existence of standard CV $t$-designs for $t \geq 2$ via contradiction.
• Establishes a link between $n$-torus $2$-designs and complete sets of MUBs in $\mathbb{C}^n$ by showing that mutually unbiased bases correspond to equal-weighted torus $2$-designs of size $n^2$.

Experimental results

Research questions

• RQ1 Do standard continuous-variable quantum state $t$-designs exist for $t \geq 2$ in infinite-dimensional Hilbert spaces?
• RQ2 Can a meaningful generalization of quantum designs be formulated for continuous-variable systems where standard designs fail?
• RQ3 Can rigged $t$-designs be constructed explicitly, and do they support practical quantum information protocols?
• RQ4 Is there a connection between torus $2$-designs and complete sets of mutually unbiased bases (MUBs) in finite-dimensional Hilbert spaces?
• RQ5 Can energy-constrained rigged designs be used to define a meaningful average fidelity for CV quantum channels and relate it to entanglement fidelity?

Key findings

• No standard continuous-variable state $t$-designs exist for $t \geq 2$ in any separable, infinite-dimensional Hilbert space, including $L^2(\mathbb{R})$.
• No continuous-variable unitary $t$-designs exist for $t \geq 2$, even when including non-Gaussian unitaries.
• Rigged $2$-designs exist and are explicitly constructed using Fock states and phase states, forming a POVM optimal for phase estimation.
• A design-based shadow tomography protocol is developed for CV states using rigged $2$-designs, enabling efficient estimation of low-degree polynomial observables.
• Energy-constrained rigged designs define an average fidelity for CV quantum channels, which is shown to be related to the CV entanglement fidelity.
• Complete sets of mutually unbiased bases in $\mathbb{C}^n$ exist if and only if there exists an equal-weighted $n$-torus $2$-design of size $n^2$ whose phases define $n$ orthonormal bases.