Entanglement Forging in Quantum Simulations

• Entanglement Forging is a hybrid quantum–classical strategy that approximates a bipartite state via a structured sum of tensor-product states, reducing qubit needs.
• It reconstructs global expectation values from local measurements, using classical post-processing to replace extensive entanglement in large systems.
• EF shows practical advantages in electronic-structure calculations, yielding significant circuit resource savings and enabling scalable quantum simulations.

Searching arXiv for recent and foundational papers on Entanglement Forging to ground the encyclopedia entry. arxiv_search(query="entanglement forging", max_results=10, sort_by="relevance") Entanglement Forging (EF) is a hybrid quantum–classical strategy for simulating a target many-body state or quantum process by replacing explicit entanglement across a bipartition with classical post-processing over smaller quantum computations. In its canonical form, EF expresses a state on a doubled Hilbert space as a short or structured sum of tensor-product states across a cut, measures local quantities on the fragments, and reconstructs global expectation values from those data. In electronic-structure settings, this often means mapping one qubit to one spatial orbital rather than one spin-orbital, thereby halving the qubit requirement by separating the $\alpha$ and $\beta$ spin sectors into distinct circuits; in broader settings, EF is a circuit-knitting or Schmidt-truncation technique that trades qubit count and circuit depth for additional measurements and classical recombination (Eddins et al., 2021).

1. Conceptual and historical framework

The defining idea of EF is to approximate or represent a bipartite state by a sum of product states across a partition $A|B$. A generic form used across the literature is

$$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$

with $s_k \ge 0$ and $\sum_k s_k^2 = 1$, or, in a variational forged form,

$$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$

The operational consequence is that one prepares only the fragment states and reconstructs the expectation values of operators that factor across the cut by classical combination of fragment correlators. In the chemistry-oriented formulation introduced for near-term hardware, this allowed ten spin-orbitals to be represented on five qubits by exploiting a spin-up versus spin-down partition and weak entanglement across that cut (Eddins et al., 2021).

The 2021 formulation established EF as a method for “doubling the size” of quantum simulations on hardware by representing a $2N$-qubit state through $N$-qubit computations and classical post-processing. Later work generalized the paradigm in several directions. “Entropy-driven entanglement forging” used subsystem entropy, entanglement structure, and symmetries to choose the bipartition and forging rank (Pérez-Obiol et al., 2024). “Entanglement Forging with generative neural network models” incorporated autoregressive neural networks to learn Schmidt-like coefficients and enable scalable sampling over many components (Huembeli et al., 2022). “Hybrid Ground-State Quantum Algorithms based on Neural Schrödinger Forging” targeted the exponential bitstring summation bottleneck of Schrödinger-picture EF by adaptively selecting the most relevant bitstrings with a generative model (Schoulepnikoff et al., 2023). Other works extended EF beyond ground-state variational simulation, including thermofield-double preparation (Faílde et al., 2023), distributed quantum computation without quantum links (Jin et al., 2024), and quantum-centric electronic-structure workflows on IBM Heron hardware (Smith et al., 11 Aug 2025).

A recurring misconception is that EF is merely qubit tapering or a symmetry reduction. The literature instead treats EF as a distinct bipartition-based reconstruction strategy: tapering removes qubits through exact conserved quantities, whereas EF retains the physical degrees of freedom but shifts the burden of inter-partition entanglement from hardware to classical post-processing (Pérez-Obiol et al., 2024).

2. Mathematical structure and expectation-value reconstruction

The mathematical core of EF is a bipartite decomposition together with an estimator for expectation values of factorized operators. In one standard formulation, the state is written as

$$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$

where $\beta$0 are computational-basis states, $\beta$1 and $\beta$2 are local unitaries, and the Schmidt coefficients satisfy $\beta$3 (Eddins et al., 2021). The associated Schrödinger-picture forging identity introduces superposition states

$\beta$4

so that off-diagonal contributions can be reconstructed from diagonal measurements on these superpositions. For a bipartite operator $\beta$5, the expectation value becomes a sum of products of $\beta$6-qubit expectation values on diagonal states and superposition states, rather than a direct $\beta$7-qubit measurement (Eddins et al., 2021).

The chemistry-specific formulation in the 2025 Heron study makes this structure explicit for spin-separated electronic Hamiltonians. There the EF ansatz is

$\beta$8

with $\beta$9 and $A|B$0 living on the $A|B$1 and $A|B$2 partitions. For tensor-product operators $A|B$3, the expectation is reconstructed from diagonal contributions and from superposition states

$A|B$4

with normalization factors

$A|B$5

This converts global expectation values into $A|B$6-only and $A|B$7-only measurements plus a controlled set of superposition measurements (Smith et al., 11 Aug 2025).

For electronic structure, the same logic appears at the reduced-density-matrix level. With spin-resolved $A|B$8-RDMs $A|B$9 and same-spin $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$0-RDMs $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$1, the energy is written as

$$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$2

Under EF, the same-spin terms are reconstructed partition-wise, while the opposite-spin contribution is bilinear in $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$3 and $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$4 and therefore requires only products of partition-wise $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$5-RDM elements rather than any $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$6-qubit cross operator (Smith et al., 11 Aug 2025).

Several EF variants differ chiefly in what is factorized. Schrödinger-picture EF forges the state and reconstructs observables from local expectations. Heisenberg-picture forging instead decomposes observables and can provide constant-overhead estimators under additional symmetry assumptions, notably when $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$7 and the problem admits the required Clifford decompositions (Eddins et al., 2021). This suggests a fundamental axis of variation in EF: state-side truncation versus operator-side decomposition.

3. Electronic-structure EF and qubit halving by spin separation

In conventional second-quantized encodings for chemistry, one qubit represents a spin-orbital, so a problem with $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$8 spatial orbitals requires $$|\psi\rangle = \sum_k s_k\, |a_k\rangle \otimes |b_k\rangle,$$9 qubits to represent the $s_k \ge 0$0 and $s_k \ge 0$1 sectors. The 2025 quantum-centric hydrogen-abstraction study implements EF by mapping one qubit to one spatial orbital and handling the $s_k \ge 0$2 and $s_k \ge 0$3 sectors on separate $s_k \ge 0$4-qubit circuits. Entanglement between these sectors is then “forged” classically from the measurement results of the two circuits (Smith et al., 11 Aug 2025).

The nonrelativistic electronic Hamiltonian is written in second quantization as

$s_k \ge 0$5

with spin-separated contributions $s_k \ge 0$6, $s_k \ge 0$7, and opposite-spin density-density terms $s_k \ge 0$8. EF is especially natural for this structure because the opposite-spin term can be assembled from products of partition-wise observables. The circuits in the Heron study use the standard Jordan–Wigner transformation, with the essential EF change being that the two spin sectors are implemented on separate spatial-orbital registers rather than on a single $s_k \ge 0$9-qubit register (Smith et al., 11 Aug 2025).

That work also broadens the expressive family of EF states beyond earlier “bitstrings + hopgates” constructions. Starting from unitary coupled-cluster doubles,

$\sum_k s_k^2 = 1$0

the authors motivate EF states of the form

$\sum_k s_k^2 = 1$1

where $\sum_k s_k^2 = 1$2 are non-orthogonal Slater determinants from a resonating Hartree–Fock optimization and the unitary doubles are implemented by a local unitary cluster Jastrow ansatz. CCSD amplitudes parameterize the LUCJ layers, while ResHF provides the non-orthogonal determinants and EF coefficients (Smith et al., 11 Aug 2025).

The original chemistry EF demonstration on water used a more compact ansatz with five hop gates and a truncated Schmidt decomposition retaining three bitstrings,

$\sum_k s_k^2 = 1$3

on a five-qubit register representing one spin sector. The same work defined the two-qubit hop gate

$\sum_k s_k^2 = 1$4

which was used as a hardware-efficient primitive for real-valued, fixed-particle-number wavefunctions (Eddins et al., 2021).

The resource advantage of spin-separated EF is concrete. After transpilation to heavy-hex topology on IBM Heron, the 2025 study reports the following average resources per circuit:

Active space	EF(ind)	EF(super)	Conventional LUCJ
$\sum_k s_k^2 = 1$5	13 qubits, 248 CZ, depth 257	13+1 qubits, 450 CZ, depth 474	26+4 qubits, 1606 CZ, depth 631
$\sum_k s_k^2 = 1$6	23 qubits, 800 CZ, depth 434	23+1 qubits, 1375 CZ, depth 834	46+6 qubits, 5167 CZ, depth 1222

The same study states that EF circuits used approximately $\sum_k s_k^2 = 1$7–$\sum_k s_k^2 = 1$8 fewer CZ gates and substantially shallower depths than the corresponding $\sum_k s_k^2 = 1$9-qubit circuits, while also yielding a consistently larger fraction of bitstrings with correct $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$0 and $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$1 across reactant, transition-state, and product geometries (Smith et al., 11 Aug 2025).

4. Measurement protocols, sampling, and integration with subspace methods

EF replaces a wide entangling circuit by a collection of smaller circuits plus classical stitching. The measurement burden therefore becomes central. In the 2025 Heron workflow, two classes of circuits are used. “EF ind” prepares the individual states $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$2 and $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$3 and measures the observables required for $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$4 and $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$5. “EF super” prepares the superposition states $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$6 and $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$7 using a controlled orbital rotation in a Hadamard-test-like construction with an ancilla initialized in

$$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$8

Measuring the ancilla in the $$|\Psi\rangle = \sum_i c_i\, |\phi_i^A\rangle \otimes |\phi_i^B\rangle.$$9 basis probabilistically prepares the required superposition states, which capture the off-diagonal terms in the global expectation values (Smith et al., 11 Aug 2025).

The same workflow integrates EF with sample-based quantum diagonalization (SQD). SQD samples computational-basis configurations from a quantum circuit, projects the Schrödinger equation into the sampled subspace, and solves

$2N$0

using Davidson because the sampled Slater determinants are orthonormal and therefore $2N$1. To handle noisy sampling and ansatz imperfections, SQD applies self-consistent configuration recovery: it postselects or repairs configurations to enforce correct particle numbers, forms randomized batches, diagonalizes each batch subspace, updates orbital occupations, and iterates to convergence (Smith et al., 11 Aug 2025).

EF contributes the joint distribution of $2N$2 and $2N$3 bitstrings in the form

$2N$4

but because the coefficients $2N$5 do not define a probability distribution, the study samples from the approximate compound distribution

$2N$6

Those paired configurations then seed the SQD recovery and diagonalization loop (Smith et al., 11 Aug 2025).

Outside chemistry, measurement and sampling are also the central challenge. In the 2021 formulation, the number of experiments required for additive error $2N$7 with $2N$8 confidence was given as

$2N$9

and the direct Schrödinger-forging sampling overhead scales as

$N$0

These formulas make explicit that EF is efficient when the bipartite entanglement is limited, since then $N$1 remains small (Eddins et al., 2021).

The generative-model EF literature addresses precisely this measurement bottleneck. The 2022 neural-network formulation uses an autoregressive neural network to model $N$2 and the ratio

$N$3

which enters the estimator for cross-register observables after a Clifford decomposition. The efficiency claim is not that EF eliminates measurement cost, but that the symmetry-restricted estimator remains polynomial in system size and measurement precision because the required conditional probabilities are sampled from single-register circuits (Huembeli et al., 2022).

5. Rank selection, entropy guidance, and neural augmentation

A central question in EF is how to choose the bipartition and how many forged terms to retain. “Entropy-driven entanglement forging” formalizes this using the Schmidt spectrum and subsystem entropy. For a pure state across $N$4 with Schmidt coefficients $N$5, the von Neumann entropy is

$N$6

and for an equipartition of $N$7 qubits the paper uses the bound

$N$8

A heuristic forging rank is then $N$9, refined by inspecting the Schmidt spectrum and symmetry-induced degeneracies (Pérez-Obiol et al., 2024).

That work shows how limited physical information can determine an effective EF structure. In the Fermi–Hubbard chain at half filling, a robust gap appears between keeping four and five Schmidt terms because one non-degenerate coefficient is followed by four symmetry-related degenerate ones; for $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$0, a $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$1 infidelity is reached by $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$2. In neutron-rich nuclear shell-model systems, proton–neutron partitions exhibit low entanglement and fivefold degenerate subleading Schmidt sectors, motivating $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$3 and, in some cases, a second forging layer that quarters the qubit count per fragment (Pérez-Obiol et al., 2024).

The resource implications are substantial but strongly regime-dependent. In the reported benchmarks, one EF cut halves the qubits per fragment, and two cuts quarter them. For $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$4, the full ADAPT-VQE circuit required more than $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$5 CNOTs by iteration $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$6, whereas one-cut EDEF reduced the maximum CNOTs per circuit by an order of magnitude and two-cut EDEF by a further order of magnitude; for $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$7, two-cut EDEF kept the largest circuit within $$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$8–$$|\psi\rangle = (U\otimes V)\sum_n \lambda_n\, |b_n\rangle \otimes |b_n\rangle,$$9 CNOTs (Pérez-Obiol et al., 2024). The same paper also identifies the principal failure mode: when the chosen cut carries strong entanglement, the fixed-rank approximation saturates and accuracy degrades unless $\beta$00 is increased.

Neural approaches address the complementary problem of selecting the relevant forged basis states without enumerating exponentially many bitstrings. “Hybrid Ground-State Quantum Algorithms based on Neural Schrödinger Forging” introduces a generative autoregressive model that learns the most important forging bitstrings and restricts the EF sum to a resource-controlled subset $\beta$01 of size $\beta$02, using the truncated estimator

$\beta$03

with truncation bias bounded by the omitted tail. The ARNN is trained with losses such as MMD,

$\beta$04

or log-cosh regression on $\beta$05, and the optimal restricted Schmidt coefficients are obtained by solving the eigenproblem

$\beta$06

for the smallest