Coherence in Property Testing: Quantum-Classical Collapses and Separations

TL;DR

It is shown that coherence alone is not enough, and a general proof cannot improve testability of any quantum property whatsoever, and connections to disentangler and quantum-to-quantum transformation lower bounds are shown.

Abstract

Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain $\{0,1\}^n$. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given $2^{n/16}$ samples, no tester can distinguish distributions of support size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$, even promised they are flat. Quantum states can be in a coherent superposition of states of $\{0,1\}^n$, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, $|φ_S \rangle=1/\sqrt{|S|}\sum_{i\in S}|i\rangle$. We show that coherence alone is not enough, Coherence limitations: Given $2^{n/16}$ copies, no tester can distinguish subset states of size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$. The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given $2^{O(n)}$ samples from a distribution and $2^{O(n)}$ communication with AM provers, no tester can estimate the support size up to factors $2^{Ω(n)}$ with probability better than $2^{-Θ(n)}$. Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.

Figures (5)

• Figure 1: Mixing to uniform measure on $X(t)$ with Down random walk starting from $X(s)$. Gray vertices on top indicate support of initial measure on $X(s)$, whereas gray vertices on the bottom indicate the support on $X(t)$.
• Figure 2: A pictorial representation of combining a quantum-to-quantum state transformation that generates suitable proofs with a property tester promised to receive copies of the input quantum state, as well as, these suitable (structured) proofs. This combination yields a property tester (depicted as the dashed enclosing box) using only copies of the input state and no proofs.
• Figure 3: We depict the relationships among property testing both in the information theoretic models (on the upper left), the computation constrained models (on the upper right), and the standard complexity classes (on the bottom) for the case of BQP, QMA, and QMA(k). Line segments from bottom to top indicate containments (i.e., the model on top can test at least all the properties its connecting bottom model can).
• Figure 4: A pictorial representation of the limitations in distinguishing support size of flat coherent quantum states (depicted on the left column) and flat classical distributions (on the right column). Dashed boxes indicate that the model fails in this task whereas a solid box indicates that the model succeeds.
• Figure :

Theorems & Definitions (78)

• Theorem 1.1: Failure of Classical Testing (Informal)
• Theorem 1.2: Failure of Testing with Coherence (Informal)
• Theorem 1.3
• Theorem 1.4: Failure of Classical Testing with Certificates (Informal)
• Theorem 1.5: Classical Certification Offer No Advantage (Informal)
• Theorem 1.6: Effective Quantum Certified Testing (Informal)
• Theorem 1.7: Informal
• Theorem 1.8: Hardness of Absolute Amplitudes Transformation (Informal)
• Corollary 1.9: Pseudorandom States
• Corollary 1.10: Pseudoentangled States