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I want to understand how the precision of promise gap on input size changes the problem's difficulty. I read the guided local Hamiltonian problem (GLHP).

Description of GLHP: We have been given a Hamiltonian and a 'good guessed' ground state. Determine if the ground state eigenvalue $\lambda_0\leq \alpha$ or $\lambda_0\geq \beta$. Where $\alpha, \beta \in \mathbb{R}$ (page 3, definition 2). In the case of the Guided Local Hamiltonian Problem (link), the hardness depends on the promise gap as follows:

  • BPP if, $\alpha - \beta = \Omega(1)$
  • BQP-complete if, $\alpha - \beta = \Omega(1/poly(n))$
  • QMA-complete if, $\alpha - \beta = \Omega(1/exp(n))$

While inspecting APPROX-QCIRCUIT-PROB (AQP), I don't find such a gap.

I want to know if the promise gap is implicit in the description of APQ.


Description of APPROX-QCIRCUIT-PROB (from wiki):

Given a description of a quantum circuit $C$ acting on $n$ qubits with $m$ gates, where $m$ is a polynomial in $n$, and each gate acts on one or two qubits, and two numbers $\alpha, \beta \in [0,1], \alpha > \beta$, distinguish between the following two cases:

  • measuring the first qubit of the state $C|0\rangle^{\otimes n}$ yields $|1\rangle$ with probability $\geq \alpha$.
  • measuring the first qubit of the state $C|0\rangle^{\otimes n}$ yields $|1\rangle$ with probability $\leq \beta$.

My attempt: I think APQ has a one-bit output, while GLHE has an n-bit output. Thus, the promise gap for AQP is not applicable.

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  • $\begingroup$ Wouldn't it be $|\alpha-\beta|$? Imagine if $\alpha=\beta$ or if $|\alpha-\beta|$ were very (exponentially) small - then even a good BQP algorithm couldn't distinguish. $\endgroup$ Commented Apr 29 at 22:15
  • $\begingroup$ Actually no need for absolute values. Just $\alpha-\beta$ because $\alpha>\beta$. $\endgroup$ Commented Apr 30 at 2:00
  • $\begingroup$ @MarkSpinelli, the key reason for my confusion is as follows: There are two sources of precision. 1. the precision on output value (say, eigenvalue estimation). 2. Precision on probabilistic guarantee, like (1/3, 2/3) or 1/2 +1/poly(n). the 2nd one is a generic feature of 'randomized' (like) computation like BPP and BQP. $\endgroup$ Commented May 1 at 6:24
  • $\begingroup$ @MarkSpinelli, I tried to pin down features 1 and 2 in AQP and GLHP problems. I can figure out feature 1 for GLHP (eigenvalue precision), while no mention of feature 2 (I think they have assumed it implicitly). While in AQP, I see feature 2 explicit (probabilistic guarantee). Since AQP is a one-bit output problem. I can't think of what does even output precision would mean. $\endgroup$ Commented May 1 at 6:30
  • $\begingroup$ @MarkSpinelli, precisely I need the following help: How to figure out features 1 and 2 in both problems. Or is there a more straightforward way to look at the problem? :) Thanks! $\endgroup$ Commented May 1 at 6:34

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For clear estimation problems such as GLHE or diagonal entries of matrix powers, the goal is to estimate a value up to some additive (or multiplicative) error. But often to prove complexity results and containment in classes such as BQP or QMA it's convenient to convert the problem to a (yes/no) decision problem. We do this by converting the additive error into a promise on the true value to be estimated.

For example, the GLHE problem as an estimation could be defined as "given a circuit to prepare good guess of the ground state of a Hamiltonian, estimate the ground state energy up to an additive error of $\alpha-\beta$." But this isn't a yes/no decision problem, so we convert it to "given a good guess as to the ground state of a Hamiltonian as a circuit to prepare this ansatz, and a promise that the true energy is greater than or equal to $\alpha$ and less than or equal to $\beta$, decide which is which."

Turning to the APPROX-QCIRCUIT-PROB, this problem is already framed as a decision problem. Thus, the gap in the above problem is already $\alpha-\beta$. If this gap gets smaller and smaller then the problem is harder and harder to solve, but as long as the gap is $\Omega(1)$ it can be amplified with repetition by applying the Chernoff bound. If the gap is too small then the Chernoff bound won't help.

But, another way to think of the APPROX-QCIRCUIT-PROB is also as an estimation problem - here, we can ask what is the true expectation of the first qubit in the circuit, up to an (additive) error of $\alpha-\beta$? Either way, the gap is $\alpha-\beta$ and as a decision problem it is not well-defined for any true expectation less than $\alpha$ and greater than $\beta$.

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