Adiabatic Quantum State Generation and Statistical Zero Knowledge

TL;DR

The paper pioneers a unified framework for quantum algorithms as quantum state generation, using adiabatic evolution to generate target ground states. It develops general tools—namely the sparse Hamiltonian lemma and the jagged adiabatic path lemma—that guarantee simulatable Hamiltonians and robust spectral gaps, establishing equivalence between adiabatic state generation and standard circuit-based state generation. By recasting SZK complete problems through quantum sampling (CQS), it connects complexity theory with quantum simulation and Markov-chain techniques, enabling Qsampling of various combinatorial and lattice distributions. The work thus links SZK, adiabatic computation, Hamiltonian simulation, and Markov-chain sampling, offering a versatile route to design new quantum algorithms and to understand the computational power of quantum state generation.

Abstract

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.

Figures (2)

• Figure 1: In the upper diagonal side of the matrix $H_m$: the row and column of $(i_b,j_b)$ are empty because the color $m$ contains the row-index and column index of $(i,j)$, and the $j_b$'th row and $i_b$'th column are empty because $m$ contains $k, ~i \bmod k,~j \bmod k$ and $i \bmod k \neq j \bmod k$. The lower diagonal side of $H_m$ is just a reflection of the upper diagonal side. It follows that ${\left\{i_b,j_b\right\}}$ is a $2 \times 2$ combinatorial block.
• Figure 2: In the left side of the drawing we see two Hamiltonians $H_1$ and $H_2$ connected by a straight line, and the spectral gaps along that line. In the right side of the drawing we see the same two Hamiltonians $H_1$ and $H_2$ connected through a jagged line that goes through a third connecting Hamiltonian $H_3$ in the middle, and the spectral gaps along that jagged path. Note that on the left the spectral gap becomes zero in the middle, while on the right it is always larger than one.

Theorems & Definitions (43)

• Definition 1
• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• Corollary 1
• Theorem 3
• Definition 2
• Claim 1
• proof