EPR* Problem: Quantum Hamiltonian Optimization

• The EPR* problem is a Hamiltonian decision and optimization challenge defined on weighted graphs using 2-local qubit interactions.
• It employs level-2 quantum moment-SoS SDP relaxations alongside a depth-1 circuit ansatz to achieve a certified 0.8395-approximation.
• The framework integrates nonlinear monogamy-of-entanglement bounds and product-to-sum control to convert local edge data into global performance guarantees.

The EPR* problem is a Hamiltonian decision and optimization problem on a weighted graph $G=(V,E,w)$ with positive edge-weights $w_{ij}>0$. Its central object is the EPR Hamiltonian

$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$

where $X_i,Y_i,Z_i$ are Pauli operators on qubit $i$. The decision version asks, for a threshold $\lambda$, whether the maximum eigenvalue $\lambda_{\max}(H(G))$ exceeds $\lambda$, while the optimization version seeks

$$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$

No polynomial-time algorithm is known; the problem lies in the class StoqMA. Recent work gives an efficient $0.8395$-approximation algorithm based on a level-2 quantum moment-SoS semidefinite relaxation, a depth-1 circuit ansatz, and a new nonlinear monogamy-of-entanglement bound on star graphs (Apte et al., 10 Dec 2025).

1. Formal statement of the problem

An instance of EPR* is specified by a graph $w_{ij}>0$0 on $w_{ij}>0$1 vertices with positive edge-weights. The associated operator $w_{ij}>0$2 is a 2-local qubit Hamiltonian obtained by summing the edge terms $w_{ij}>0$3. The optimization target is the spectral quantity $w_{ij}>0$4, and the decision problem asks whether this value is larger than a prescribed threshold $w_{ij}>0$5.

An $w_{ij}>0$6-approximation algorithm for EPR* outputs in polynomial time a number $w_{ij}>0$7 satisfying

$w_{ij}>0$8

The paper establishes such a guarantee with $w_{ij}>0$9, thereby improving the approximation ratio for the EPR Hamiltonian (Apte et al., 10 Dec 2025).

Quantity	Definition	Role
$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$0	$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$1	Objective Hamiltonian
$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$2	$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$3	Edge interaction
$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$4	$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$5	Optimum value
$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$6	Polynomial-time output	Approximate value

2. Semidefinite relaxation and edge parameters

The approximation framework begins with an upper bound on $$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$7 obtained from the level-2 quantum moment-SoS SDP relaxation. Its dual yields edge-dependent quantities

$$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$8

where $$H(G)\;=\;\sum_{(i,j)\in E} w_{ij}\,h_{ij}, \qquad h_{ij}\;=\;\tfrac12\bigl(I_iI_j+X_iX_j-\,Y_iY_j+Z_iZ_j\bigr),$$9 is the pseudo-expectation produced by the SDP. These variables compress the relaxation into edge-local surrogates that subsequently parameterize the circuit ansatz.

From the dual data one forms

$X_i,Y_i,Z_i$0

and the relaxation guarantees

$X_i,Y_i,Z_i$1

This upper bound is the benchmark against which the constructive lower bound is compared. The overall strategy is therefore not merely to produce a candidate quantum state, but to prove that its value is a uniform fraction of an efficiently computable SDP upper bound.

3. Depth-1 circuit ansatz and certified lower bound

After solving the level-2 SDP, the algorithm computes angles $X_i,Y_i,Z_i$2 via a fixed scalar function $X_i,Y_i,Z_i$3. It then prepares the $X_i,Y_i,Z_i$4-qubit state

$X_i,Y_i,Z_i$5

This is a depth-1 product ansatz over edges. The Hamiltonian value $X_i,Y_i,Z_i$6 can be estimated by measurement, but the analysis also furnishes a deterministic classical lower bound (Apte et al., 10 Dec 2025).

That lower bound is

$X_i,Y_i,Z_i$7

where

$X_i,Y_i,Z_i$8

Since $X_i,Y_i,Z_i$9, a bound of the form $i$0 immediately yields an $i$1-approximation. The analytical problem is thus reduced to controlling products of cosines of edge angles derived from the SDP data.

4. Nonlinear monogamy on stars and the $i$2 parameterization

The key technical ingredient is a nonlinear monogamy-of-entanglement bound on star graphs. For any vertex $i$3 of degree $i$4 and any neighbor $i$5, if

$i$6

then

$i$7

The same bound also holds for the level-2 SDP variables $i$8. In positive-part form, with $i$9, this yields the corollary

$\lambda$0

where

$\lambda$1

This star inequality is what permits local edge information to constrain all other adjacent edges (Apte et al., 10 Dec 2025).

To exploit the bound, the paper introduces scalar functions

$\lambda$2

with $\lambda$3 increasing, $\lambda$4, and satisfying the product-to-sum condition

$\lambda$5

for nonnegative $\lambda$6 with $\lambda$7. One then defines

$\lambda$8

Concretely, the chosen constants are

$\lambda$9

The function $\lambda_{\max}(H(G))$0 is piecewise linear through

$\lambda_{\max}(H(G))$1

and

$\lambda_{\max}(H(G))$2

5. Worst-case-edge analysis and the $\lambda_{\max}(H(G))$3 guarantee

The proof reduces the global approximation ratio to a one-edge minimization: $\lambda_{\max}(H(G))$4 The paper then analyzes three exhaustive cases. In the first case, all incident $\lambda_{\max}(H(G))$5-values satisfy $\lambda_{\max}(H(G))$6, giving

$\lambda_{\max}(H(G))$7

and hence

$\lambda_{\max}(H(G))$8

In the second case, the edge value itself satisfies $\lambda_{\max}(H(G))$9. Then

$\lambda$0

which yields

$\lambda$1

In the third case, there is a neighbor $\lambda$2 with $\lambda$3. By bipartiteness and the monogamy bound, all other adjacent $\lambda$4-values are then at most $\lambda$5, and a refined estimate produces a third one-dimensional ratio $\lambda$6. Claim 3.9 asserts

$\lambda$7

A careful numerical, but rigorously checked, evaluation shows that each of these three minima is at least $\lambda$8. This establishes the certified $\lambda$9-approximation ratio (Apte et al., 10 Dec 2025).

The proof architecture combines three ingredients: the nonlinear star monogamy bound, the product-to-sum control built into $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$0, and the reduction to three one-dimensional minimizations. The full technical proofs appear in Sections 4–6 of the paper, and the underlying control problem is to bound

$$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$1

in terms of sums of positive parts $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$2.

6. Limits of the current framework

The same work also proves that the present method is close to its internal limit. First, any analysis that both chooses $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$3 and lower-bounds $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$4 by the minimum over edges cannot exceed

$$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$5

This no-go result shows that the certified $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$6 ratio is essentially optimal within that proof template (Apte et al., 10 Dec 2025).

Second, the depth-1 product ansatz

$$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$7

cannot surpass

$$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$8

on the 4-cycle, even with fully global angle selection as a function of all $$\lambda_{\max}(H(G))\;=\;\max_{\ket\psi}\bra\psi\,H(G)\,\ket\psi.$$9. The paper therefore concludes that any route to approximation ratios above $0.8395$0 must use deeper circuits or completely new ideas. In the paper’s own summary, current SDP-plus-depth-1-circuit methods cannot achieve substantially better approximation ratios, and further progress will require fundamentally new techniques.

Within this landscape, EPR* occupies a precise niche: it is a graph-structured quantum Hamiltonian optimization problem for which semidefinite relaxations, edge-wise pseudo-moment data, shallow entangling circuits, and monogamy constraints can be assembled into a provable approximation algorithm. The current frontier is not merely the $0.8395$1 factor itself, but the delineation of which parts of that factor arise from the Hamiltonian’s geometry and which arise from the limitations of the analytic and variational framework presently available.