arXiv subjects

Yeongwoo Hwang

Publications and source records attributed to Yeongwoo Hwang.

5 recordsLinked to original sources

arXiv2026

A quantum oracle separation between QMA(2) and QMA

We find a quantum oracle relative to which $\mathsf{QMA} \neq \mathsf{QMA}(2)$. As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every $ε+δ<1$, any $(ε,δ)$-disentangler requires input size exponential in the number of output qubits. Our proof combines the unitarily invariant polynomial method of She and Yuen (ITCS '23) with a new construction based on the symmetric and antisymmetric subspace projectors, reducing the $\mathsf{QMA}$ lower bound to the approximate degree of $\mathrm{OR}$.

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arXiv2026

Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling

Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature. Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH $H$ on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian $H^{(c)}$. We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows. - If $H$ is a 2-local qudit CLH, then $H^{(c)}$ is a 2-local qudit classical Hamiltonian. - If $H$ is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then $H^{(c)}$ is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in $O(n^2 poly(log n))$ time. - If $H$ is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then $H^{(c)}$ is a constant-local classical Hamiltonian.

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arXiv2025

On the complexity of unique quantum witnesses and quantum approximate counting

We study the long-standing open question on the power of unique witnesses in quantum protocols, which asks if $\textsf{UniqueQMA}$, a variant of $\textsf{QMA}$ whose accepting witness space is 1-dimensional, contains $\mathsf{QMA}$ under quantum reductions. This work rules out any black-box reduction from $\mathsf{QMA}$ to $\mathsf{UniqueQMA}$ by showing a quantum oracle separation between $\mathsf{BQP}^\mathsf{UniqueQMA}$ and $\mathsf{QMA}$. This provides a contrast to the classical case, where the Valiant-Vazirani theorem shows a black-box randomized reduction from $\mathsf{UniqueNP}$ to $\mathsf{NP}$, and suggests the need for studying the structure of the ground space of local Hamiltonians in distilling a potential unique witness. Via similar techniques, we show, relative to a quantum oracle, that $\mathsf{QMA}^\mathsf{QMA}$ cannot decide quantum approximate counting, ruling out a quantum analogue of Stockmeyer's algorithm in the black-box setting. We then ask a natural question; what structural properties of the local Hamiltonian problem can we exploit? We introduce a physically motivated candidate by showing that the ground energy of local Hamiltonians that satisfy a computational variant of the eigenstate thermalization hypothesis (ETH) can be estimated through a $\mathsf{UniqueQMA}$ protocol. Our protocol can be viewed as a quantum expander test in a low energy subspace of the Hamiltonian and verifies a unique entangled state across two copies of the subspace. This allows us to conclude that if $\mathsf{UniqueQMA}$ is not equivalent to $\mathsf{QMA}$, then $\mathsf{QMA}$-hard Hamiltonians must violate ETH under adversarial perturbations. This also serves as evidence that chaotic local Hamiltonians, such as the SYK model may be computationally simpler than general local Hamiltonians.

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arXiv2025

Commuting Local Hamiltonians Beyond 2D

Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the existence of area laws. Although they are a simplified model of quantum computation, the status of the commuting local Hamiltonian problem remains largely unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit completely classical verifiers. Despite intense work, the largest class of commuting local Hamiltonians we can place in NP are those on a square lattice, where each lattice site is a qutrit. Even worse, many of the techniques used to analyze these problems rely heavily on the geometry of the square lattice and the properties of the numbers 2 and 3 as local dimensions. In this work, we present a new technique to analyze the complexity of various families of commuting local Hamiltonians: guided reductions. Intuitively, these are a generalization of typical reduction where the prover provides a guide so that the verifier can construct a simpler Hamiltonian. The core of our reduction is a new rounding technique based on a combination of Jordan's Lemma and the Structure Lemma. Our rounding technique is much more flexible than previous work, and allows us to show that a larger family of commuting local Hamiltonians is in NP, albiet with the restriction that all terms are rank-1. Specifically, we prove the following two results: 1. Commuting local Hamiltonians in 2D that are rank-1 are contained in NP, independent of the qudit dimension. Note that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality. 2. We prove that rank-1, 3D commuting Hamiltonians with qudits on edges are in NP. To our knowledge this is the first time a family of 3D commuting local Hamiltonians has been contained in NP.

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arXiv2022

Unique Games hardness of Quantum Max-Cut, and a conjectured vector-valued Borell's inequality

The Gaussian noise stability of a function $f:\mathbb{R}^n \to \{-1, 1\}$ is the expected value of $f(\boldsymbol{x}) \cdot f(\boldsymbol{y})$ over $ρ$-correlated Gaussian random variables $\boldsymbol{x}$ and $\boldsymbol{y}$. Borell's inequality states that for $-1 \leq ρ\leq 0$, this is minimized by the halfspace $f(x) = \mathrm{sign}(x_1)$. In this work, we generalize this result to hold for functions $f:\mathbb{R}^n \to S^{k-1}$ which output $k$-dimensional unit vectors. Our main conjecture, which we call the $\textit{vector-valued Borell's inequality}$, asserts that the expected value of $\langle f(\boldsymbol{x}), f(\boldsymbol{y})\rangle$ is minimized by the function $f(x) = x_{\leq k} / \Vert x_{\leq k} \Vert$, where $x_{\leq k} = (x_1, \ldots, x_k)$. We give several pieces of evidence in favor of this conjecture, including a proof that it does indeed hold in the special case of $n = k$. As an application of this conjecture, we show that it implies several hardness of approximation results for a special case of the local Hamiltonian problem related to the anti-ferromagnetic Heisenberg model known as Quantum Max-Cut. This can be viewed as a natural quantum analogue of the classical Max-Cut problem and has been proposed as a useful testbed for developing algorithms. We show the following, assuming our conjecture: (1) The integrality gap of the basic SDP is $0.498$, matching an existing rounding algorithm. Combined with existing results, this shows that the basic SDP does not achieve the optimal approximation ratio. (2) It is Unique Games-hard (UG-hard) to compute a $(0.956+\varepsilon)$-approximation to the value of the best product state, matching an existing approximation algorithm. (3) It is UG-hard to compute a $(0.956+\varepsilon)$-approximation to the value of the best (possibly entangled) state.

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