Amplitude Amplification for Gutzwiller Projection

• AAGP is a quantum-state preparation approach that replaces measurement-based postselection with coherent amplitude amplification to enhance the Gutzwiller-allowed subspace.
• The method employs a Grover-style fixed-point search, reducing the query complexity from O(1/𝒲) to O(1/√𝒲) while managing the cost of repeated unitary operations like U_BCS.
• AAGP facilitates scalable preparation of projected BCS and RVB states, though it requires precise phase control and still exhibits exponential scaling in worst-case scenarios.

Searching arXiv for the specified papers and closely related work on Gutzwiller projection and amplitude amplification. Amplitude amplification for Gutzwiller projection (AAGP) is a family of quantum-state-preparation procedures that replaces measurement-based postselection of a non-unitary Gutzwiller projection by coherent amplitude amplification of the branch that already lies in the Gutzwiller-allowed subspace. In the literature assembled here, the idea appears first as a natural extension of an ancilla-based digital implementation of the Gutzwiller wave function for the 1D Fermi–Hubbard model, and then as an explicit scalable protocol for preparing Gutzwiller-projected Bardeen–Cooper–Schrieffer (BCS), or resonating valence bond (RVB), states for quantum simulation. Its defining feature is the replacement of an average cost proportional to the inverse projected-state weight by a Grover-style or fixed-point cost proportional to its inverse square root, while preserving the same physically motivated target state (Murta et al., 2021, Kang et al., 5 Jun 2026).

1. Gutzwiller projection as a quantum-state preparation problem

The Gutzwiller construction suppresses or removes doubly occupied sites in fermionic many-body states. For the 1D Fermi–Hubbard model with open boundary conditions at half filling, the Hamiltonian is

$$\hat{\mathcal H} = -t \sum_{i=1}^{N} \sum_{\sigma=\uparrow,\downarrow} \left(\hat a_{i,\sigma}^\dagger \hat a_{i+1,\sigma} + \text{H.c.}\right) + U \sum_{i=1}^{N} \hat n_{i,\uparrow} \hat n_{i,\downarrow},$$

and the Gutzwiller wave function is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$

where $g\in[0,1]$ is the Gutzwiller variational parameter and $\ket{\psi_0}$ is the noninteracting ground state. If a basis state has $n$ doubly occupied sites, its amplitude is multiplied by $(1-g)^n$. The limits are $U=0\Rightarrow g=0$, for which $\ket{\psi_G}=\ket{\psi_0}$, and $U\to\infty\Rightarrow g\to 1$, corresponding to full projection against double occupancy (Murta et al., 2021).

For projected BCS states, the strict no-double-occupancy projector is

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$

and the target RVB state is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$0

with normalization constant $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$1. In the $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$2–$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$3 model, this restriction is exact: the physical Hilbert space has no double occupancy, so the Gutzwiller projection is not merely variational but enforces the physical subspace itself (Kang et al., 5 Jun 2026).

A central quantity for AAGP is the projected-state weight

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$4

This quantity is simultaneously the success probability of naive projection by postselection and the amplitude parameter that determines the number of amplitude-amplification steps needed in coherent schemes (Kang et al., 5 Jun 2026).

2. Measurement-based origin of AAGP

The digital implementation that motivates AAGP prepares the noninteracting state $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$5 and then applies the non-unitary Gutzwiller projector by embedding it into a larger unitary acting on system qubits plus ancillas. Under the Jordan–Wigner mapping, each spin-orbital is encoded in one qubit, so the physical system uses $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$6 qubits. The noninteracting state is prepared in the single-particle eigenbasis and mapped to the site basis with Givens rotations; at half filling the total CNOT count for this preparation is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$7

with depth

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$8

These costs are polynomial in $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$9 (Murta et al., 2021).

The local Gutzwiller gadget uses, for each site $g\in[0,1]$0, the two system qubits $g\in[0,1]$1 and one ancilla $g\in[0,1]$2. The ancilla is initialized in $g\in[0,1]$3, and a controlled-controlled-$g\in[0,1]$4 gate is applied only if the site is doubly occupied. The single-qubit unitary is

$g\in[0,1]$5

with

$g\in[0,1]$6

Conditioned on measuring ancilla outcome $g\in[0,1]$7, the system undergoes exactly the local non-unitary map

$g\in[0,1]$8

For the full lattice, one ancilla per site is used, all local gadgets are applied, and postselection on the global ancilla outcome $g\in[0,1]$9 yields $\ket{\psi_0}$0 on the system register (Murta et al., 2021).

This construction makes the bottleneck mathematically explicit. If $\ket{\psi_0}$1 denotes the global unitary acting on ancillas plus system, the Kraus operator associated with ancilla outcome $\ket{\psi_0}$2 is

$\ket{\psi_0}$3

The success probability is therefore

$\ket{\psi_0}$4

Expanded over basis states with exactly $\ket{\psi_0}$5 doubly occupied sites,

$\ket{\psi_0}$6

At $\ket{\psi_0}$7, $\ket{\psi_0}$8, but for fixed $\ket{\psi_0}$9 and increasing $n$0, the average number of repetitions $n$1 grows exponentially with $n$2, approximately as

$n$3

Illustrative values from the simulations are modest at weak coupling but severe at strong coupling: for $n$4, the average repetitions are $n$5 at $n$6 and $n$7 at $n$8; for $n$9, they are $(1-g)^n$0 at $(1-g)^n$1 and $(1-g)^n$2 at $(1-g)^n$3 (Murta et al., 2021).

3. Amplitude-amplified formulation

The structural reason AAGP is possible is that the postselected Gutzwiller routine already provides the standard ingredients of amplitude amplification. Let $(1-g)^n$4 prepare $(1-g)^n$5 from a computational-basis reference state, and let $(1-g)^n$6 denote the coherent ancilla-plus-system projection circuit. Then a global preparation unitary $(1-g)^n$7 can be defined by

$(1-g)^n$8

where

$(1-g)^n$9

The “good” subspace is the subspace in which all ancillas are $U=0\Rightarrow g=0$0,

$U=0\Rightarrow g=0$1

and its projector is

$U=0\Rightarrow g=0$2

A corresponding phase-marking oracle is

$U=0\Rightarrow g=0$3

The reflection about the known initial computational-basis state is

$U=0\Rightarrow g=0$4

Standard amplitude amplification then uses

$U=0\Rightarrow g=0$5

If the initial success probability is $U=0\Rightarrow g=0$6, iterating $U=0\Rightarrow g=0$7 boosts the good component from cost $U=0\Rightarrow g=0$8 under repeat-until-success postselection to $U=0\Rightarrow g=0$9 coherent iterations in the small-probability regime (Murta et al., 2021).

This is the core idea behind AAGP: the expensive object is not the measurement itself but repeated use of the projection machinery. Amplitude amplification reduces the number of such uses quadratically. The 2021 study explicitly notes that combining the non-deterministic scheme with quantum amplitude amplification “could lead to a trade-off between the number of repetitions and the circuit depth,” but does not design the full scheme in detail (Murta et al., 2021).

A recurring misconception is that this change makes Gutzwiller projection polynomial-time in regimes where postselection is exponential. The available analyses do not support that conclusion. If $\ket{\psi_G}=\ket{\psi_0}$0, amplitude amplification improves this to $\ket{\psi_G}=\ket{\psi_0}$1 at the level of query complexity, which is a quadratic gain in the success amplitude but not a change from exponential to polynomial scaling (Murta et al., 2021, Kang et al., 5 Jun 2026).

4. Non-Boolean amplitude amplification as a soft-projector perspective

A distinct but closely related framework generalizes amplitude amplification from Boolean phase-flip oracles to non-Boolean phase oracles

$\ket{\psi_G}=\ket{\psi_0}$2

where the phase $\ket{\psi_G}=\ket{\psi_0}$3 is real-valued and need not be restricted to $\ket{\psi_G}=\ket{\psi_0}$4 or $\ket{\psi_G}=\ket{\psi_0}$5. In the two-register construction, one defines

$\ket{\psi_G}=\ket{\psi_0}$6

and

$\ket{\psi_G}=\ket{\psi_0}$7

The key scalar parameter is

$\ket{\psi_G}=\ket{\psi_0}$8

After $\ket{\psi_G}=\ket{\psi_0}$9 iterations, the system probability distribution obeys

$U\to\infty\Rightarrow g\to 1$0

with

$U\to\infty\Rightarrow g\to 1$1

When $U\to\infty\Rightarrow g\to 1$2, states with lower $U\to\infty\Rightarrow g\to 1$3 are amplified at the expense of states with higher $U\to\infty\Rightarrow g\to 1$4 (Shyamsundar, 2021).

This furnishes a soft-projector viewpoint on Gutzwiller-type filtering. A plausible implication is that one may encode a graded penalty for double occupancy or other constraint violations into $U\to\infty\Rightarrow g\to 1$5, choose the mapping so that more desirable configurations correspond to lower $U\to\infty\Rightarrow g\to 1$6, and then use non-Boolean amplitude amplification as a controlled, basis-diagonal bias toward the Gutzwiller-allowed sector. In that interpretation, the linear update

$U\to\infty\Rightarrow g\to 1$7

acts as a soft rather than strictly binary projector (Shyamsundar, 2021).

This perspective also clarifies the distinction between hard and soft Gutzwiller constraints. A strict projector such as

$U\to\infty\Rightarrow g\to 1$8

is a step function in occupation space, whereas the non-Boolean scheme induces a controlled redistribution of weight that is affine in $U\to\infty\Rightarrow g\to 1$9. This suggests utility for finite-penalty or graded-constraint settings, but it also indicates that non-Boolean amplitude amplification is not, by itself, an exact replacement for a sharp Gutzwiller projector unless the phase encoding is chosen in a special way (Shyamsundar, 2021).

5. Explicit scalable AAGP for projected BCS and RVB states

The explicit 2026 realization of AAGP targets arbitrary BCS states and their Gutzwiller-projected counterparts. The BCS state is prepared from a Bogoliubov–de Gennes Hamiltonian via Bloch–Messiah decomposition and Givens rotations, yielding a unitary

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$0

The total two-qubit gate count for $$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$1 is

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$2

and the circuit depth is

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$3

With $$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$4-rotations approximated to precision $$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$5, the $$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$6-count scales as

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$7

for $$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$8 (Kang et al., 5 Jun 2026).

The amplitude-amplification input/output relation is written as

$$\mathcal{P}_{\rm G} = \prod_{i=1}^{N_s} (1 - n_{i\uparrow} n_{i\downarrow}),$$9

The two phase operations required by AAGP are

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$00

and

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$01

Direct implementation of a phase oracle on the full many-body RVB state would be as hard as the target problem itself, but within the two-dimensional subspace spanned by $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$02, $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$03 acts identically to a phase rotation about $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$04. This is the crucial reduction that makes AAGP operational (Kang et al., 5 Jun 2026).

The protocol uses a fixed-point search construction. At odd steps one applies

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$05

and at even steps

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$06

The overall ordered product after $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$07 steps is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$08

with $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$09 odd. The phases are chosen as

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$10

where $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$11 satisfies

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$12

with $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$13 the Chebyshev polynomial of the first kind and $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$14 the target failure tolerance (Kang et al., 5 Jun 2026).

The fixed-point guarantee is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$15

for small $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$16 and large $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$17. Accordingly, the number of projection queries scales as

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$18

whereas measurement-based postselection requires

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$19

This is the explicit AAGP manifestation of the quadratic reduction in projection queries (Kang et al., 5 Jun 2026).

Method	Query scaling	Representative 100-site benchmark
Postselection	$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$20	$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$21
AAGP	$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$22	$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$23
Projected-state weight	$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$24	At $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$25, $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$26

For projected BCS states optimized for the square-lattice $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$27–$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$28 model, the projected-state weight decays exponentially with system size,

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$29

At $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$30, the fitted decay constant is $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$31, and for $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$32, $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$33. The corresponding reduction in required projection queries is about seven orders of magnitude (Kang et al., 5 Jun 2026).

6. Resource profile, applications, and limitations

The dominant asymptotic cost of the 2026 AAGP protocol comes from repeated applications of $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$34 and $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$35, not from the Gutzwiller oracle itself. The oracle $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$36 is implemented by computing double-occupancy flags into $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$37 site ancillas, mapping the all-zero flag pattern onto a global ancilla with multi-controlled logic, applying a phase rotation, and uncomputing. The reflection $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$38 is implemented analogously with a multi-controlled NOT conditioned on the vacuum configuration. Their costs are $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$39 in $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$40-gates and are subleading relative to the $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$41 cost of $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$42 (Kang et al., 5 Jun 2026).

The resulting total fault-tolerant scaling is

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$43

whereas postselection gives

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$44

This is the precise sense in which AAGP is fault-tolerant-friendly: the dominant dependence on the projected-state weight is improved from $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$45 to $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$46 (Kang et al., 5 Jun 2026).

The principal applications are as input-state preparation for long-depth quantum algorithms. The 2021 work discusses the Gutzwiller state as an improved initial state for quantum phase estimation, quantum imaginary time evolution, and the variational quantum eigensolver. There, the qualitative benefit is a larger overlap with the exact ground state than simpler initial guesses, but the naive postselection overhead can erase that benefit. AAGP directly addresses that bottleneck by reducing the number of times the projection machinery must be invoked coherently (Murta et al., 2021).

Several limitations are explicit in the literature. First, even with amplitude amplification, the cost remains exponential in $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$47 when $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$48 decays exponentially; AAGP halves the exponent but does not remove it (Kang et al., 5 Jun 2026). Second, the fixed-point construction assumes an estimate of $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$49 to set $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$50 and the phases $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$51; modest errors are reported as tolerable, but strong misestimation can degrade performance (Kang et al., 5 Jun 2026). Third, the strict no-double-occupancy projector is exact for the $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$52–$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$53 limit but not for finite-$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$54 Hubbard models, where physically relevant states include some double occupancy. Direct transplantation of the strict AAGP oracle to finite-$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$55 settings would therefore be inappropriate without modification (Kang et al., 5 Jun 2026).

A final misconception is that AAGP is synonymous with one unique circuit identity. The present literature supports a narrower statement: AAGP denotes amplitude amplification applied to Gutzwiller projection, either as a coherent wrapping of ancilla-based postselected projection or, more concretely, as the fixed-point projected-BCS algorithm built from $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$56, $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$57, and $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$58. This suggests a broader design space that includes strict projectors, soft phase-based filters, and finite-$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$59 extensions such as

$$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$60

where $$\ket{\psi_G} = \prod_{i = 1}^{N} \bigl(\mathbb{1} - g\, \hat n_{i,\uparrow} \hat n_{i,\downarrow}\bigr)\, \ket{\psi_0} = \hat P_G(g)\ket{\psi_0},$$61 reintroduces controlled double occupancy perturbatively after the Gutzwiller core has been prepared (Shyamsundar, 2021, Kang et al., 5 Jun 2026).