Quantum Paldus Transform (QPT)

• Quantum Paldus Transform is an isometry that remaps the fermionic Fock basis into a symmetry-adapted Gelfand–Tsetlin basis, enabling block-diagonalization into (N,S) sectors.
• It facilitates an exact matrix product state (MPS) representation of Complete Active Space (CAS) wavefunctions with polynomial bond dimension and efficient quantum circuit implementation.
• Utilizing U(d)×SU(2) Howe duality, QPT enhances spin-free Hamiltonian simulations, supports CSF preparation, and provides resource-efficient schemes in quantum chemistry.

The Quantum Paldus Transform (QPT) is an isometry $U_{\rm QPT}$ that maps the occupation-number basis of a fermionic Fock space with $2d$ spin-orbitals to a symmetry-adapted basis of Gelfand–Tsetlin (GT) states labelled by total particle number $N$, total spin $S$, spin projection $M$, and $U(d)$ GT data. In the Unitary Group Approach (UGA), it realises antisymmetric $U(d)\times SU(2)$ Howe duality and yields a basis that block-diagonalises spin-free Hamiltonians into irreducible $(N,S)$ sectors. In the complete active space (CAS) setting, the same transform is specialised to fixed $(N,S,M)$ sectors and to Shavitt-graph walks $\delta_i\in\{0,1,2,3\}$, which enables an exact matrix product state (MPS) representation of CAS wavefunctions with bond dimension $2d$0, together with second- and first-quantized state-preparation routines whose complexity grows as $2d$1 (Burkat et al., 10 Jun 2025, Jnane, 17 Jun 2026).

1. Definition of the transform

For $2d$2 spatial orbitals with spin states $2d$3, the Fock basis is

$2d$4

or equivalently

$2d$5

The QPT is the isometry

$2d$6

with

$2d$7

and inverse map

$2d$8

In the CAS-sector formulation, the corresponding map is written

$2d$9

where $N$0 records the Shavitt-graph step at orbital $N$1. The GT basis states admit the expansion

$N$2

with $N$3 for $N$4, $N$5 for $N$6 or $N$7, and $N$8 for $N$9. Once $S$0 are fixed,

$S$1

so on a fixed $S$2 sector the transform acts as a permutation/isometry between the Fock states and the $S$3-register. No simple factorization in terms of one- and two-fermion operators is given; the construction is specified through the GT expansion and the rule that $S$4 “reads off” $S$5 (Burkat et al., 10 Jun 2025, Jnane, 17 Jun 2026).

2. Representation-theoretic structure

The QPT is grounded in the decomposition of the fermionic Fock space

$S$6

Here the commuting Lie-algebra actions are generated by

$S$7

with

$S$8

and $S$9. This commuting structure is the algebraic content of Paldus duality and underlies the block structure in $M$0 (Burkat et al., 10 Jun 2025).

The GT basis is obtained by multiplicity-free recursive branching $M$1. The resulting states $M$2 are in one-to-one correspondence with walks on a Shavitt graph, or equivalently with triangular GT patterns. In the CAS construction, this graph-theoretic organisation is described as a transformation from the Fock basis to a “friendlier symmetry-adapted basis.” A plausible implication is that the efficiency of the later MPS construction is not an accident of tensor-network parametrisation alone; it is tied to the combinatorial constraints imposed by the symmetry-adapted branching data (Jnane, 17 Jun 2026).

3. Quantum-circuit realisation

A direct circuit implementation of the QPT maintains four registers: $M$3, $M$4, $M$5, and $M$6, with $M$7, $M$8, $M$9, and $U(d)$0 qubits respectively. For an input $U(d)$1, the circuit applies $U(d)$2 successive Clebsch–Gordan subroutines, each processing the pair $U(d)$3. At step $U(d)$4, it updates $U(d)$5, applies a multiplexed Givens rotation $U(d)$6 controlled by $U(d)$7, outputs two new bits $U(d)$8 encoding one of the four couplings $U(d)$9, then updates $U(d)\times SU(2)$0 and $U(d)\times SU(2)$1 (Burkat et al., 10 Jun 2025).

The elementary subroutines are expressed in fault-tolerant terms. Controlled incrementers on $U(d)\times SU(2)$2-bit registers cost $U(d)\times SU(2)$3 Toffolis using conditionally clean qubits, or $U(d)\times SU(2)$4 Toffolis with five dirty qubits. Multiplexers $U(d)\times SU(2)$5 can be implemented in $U(d)\times SU(2)$6 Toffolis by unary iteration. Controlled Givens rotations $U(d)\times SU(2)$7 are approximated via data-lookup-based or phase-gradient primitives with cost $U(d)\times SU(2)$8 Toffolis per layer, or via unary iteration in $U(d)\times SU(2)$9 Toffolis. Since at the $(N,S)$0-th step there are $(N,S)$1 distinct $(N,S)$2 rotation angles, $(N,S)$3, while the total incrementer cost is $(N,S)$4. The resulting QPT compilation has $(N,S)$5 Toffoli complexity, with $(N,S)$6 sequential Clebsch–Gordan layers and layer depth $(N,S)$7. The paper also lists three constant-factor optimisations: measurement-based uncomputation, phase-gradient reuse, and multi-index QROM or clean/dirty $(N,S)$8/$(N,S)$9, the last of which can reduce constants by $(N,S,M)$0 (Burkat et al., 10 Jun 2025).

4. CAS wavefunctions in the Paldus basis

A CAS wavefunction in the UGA basis is written as

$(N,S,M)$1

where $(N,S,M)$2 is the set of valid walks on the Shavitt graph consistent with total electron number $(N,S,M)$3 and total spin $(N,S,M)$4. The central observation of the 2026 construction is that these amplitudes can be represented exactly as an MPS over the physical alphabet $(N,S,M)$5 (Jnane, 17 Jun 2026).

The bond index at site $(N,S,M)$6 is defined as $(N,S,M)$7, where $(N,S,M)$8 counts double occupations in orbitals $(N,S,M)$9, $\delta_i\in\{0,1,2,3\}$0 counts single occupations, and $\delta_i\in\{0,1,2,3\}$1 counts empties, subject to

$\delta_i\in\{0,1,2,3\}$2

With $\delta_i\in\{0,1,2,3\}$3 and $\delta_i\in\{0,1,2,3\}$4 fixed by $\delta_i\in\{0,1,2,3\}$5, the amplitudes satisfy

$\delta_i\in\{0,1,2,3\}$6

where each $\delta_i\in\{0,1,2,3\}$7 is a very sparse indicator matrix that enforces $\delta_i\in\{0,1,2,3\}$8 and multiplies by the scalar $\delta_i\in\{0,1,2,3\}$9 if that path is in the support of the CAS. Because $2d$00, the number of possible bond states obeys

$2d$01

and therefore

$2d$02

The outcome is an exact MPS with bond dimension $2d$03. This directly supports the claim that CAS states expanded in the Paldus basis admit an efficient classical representation, despite the statement that encoding a CAS state classically is traditionally believed to be intractable for chemically relevant systems (Jnane, 17 Jun 2026).

5. CAS-state preparation and resource estimates

The CAS-state preparation protocol has two stages. First, the MPS over the $2d$04-alphabet is loaded into the registers $2d$05. Second, the inverse transform $2d$06 converts that symmetry-adapted representation back into the standard second-quantized Fock basis. The 2026 paper gives an explicit loading routine and combines it with the inverse-QPT circuit complexity inherited from the earlier QPT construction (Jnane, 17 Jun 2026).

At each site $2d$07, the loading routine carries forward a bond register $2d$08, initialises two qubits $2d$09 for the physical $2d$10 register, and a temporary QROM register $2d$11 for angle loading. The unitary $2d$12 acts as

$2d$13

Its implementation consists of four steps: QROAM; a controlled $2d$14, $2d$15 tree preparing $2d$16 on the physical two-qubit register; inverse QROAM; and controlled adders updating $2d$17 conditioned on $2d$18. For fixed $2d$19, the resource counts per $2d$20 are $2d$21 Toffolis for QROAM and its inverse, $2d$22 for the $2d$23-bit rotations, and $2d$24 for the adders. With $2d$25, $2d$26, and $2d$27, the dominant cost per site is $2d$28 Toffolis, giving $2d$29 Toffolis and depth over all $2d$30 sites.

Procedure	Resource estimate	Role
MPS load	$2d$31 Toffolis, depth $2d$32	Prepare $2d$33
Inverse QPT $2d$34	$2d$35 Toffolis, depth $2d$36	Recover the second-quantized GT/Fock state
Second-quantized CAS preparation	$2d$37 total	Combine MPS load and inverse QPT
First-quantized CAS preparation	$2d$38 total	Conversion cost $2d$39 is subleading

For the inverse transform, the CAS paper cites a detailed circuit by Burkat et al. with $2d$40 Toffoli count and $2d$41 circuit depth. Consequently, the total second-quantized cost is

$2d$42

which the paper describes as, to the best of its knowledge, an exponential improvement over the state of the art for CAS-state preparation. The first-quantized version has the same asymptotic complexity because the determinant-to-antisymmetrized-list conversion costs $2d$43, which is subleading (Jnane, 17 Jun 2026).

6. Applications, scope, and conceptual clarifications

The 2025 QPT paper identifies four principal applications. First, spin-free second-quantized Hamiltonians,

$2d$44

lie in the universal-enveloping algebra of $2d$45, so the Paldus basis block-diagonalises them into irreducible $2d$46 sectors and achieves maximal sparsity. The Fermi–Hubbard Hamiltonian is given as an explicit example that acts irreducibly within each $2d$47 block. Second, the transform supports efficient preparation of configuration-state functions (CSFs): one prepares $2d$48 and applies $2d$49, or, conversely, applies $2d$50, measures $2d$51, and post-selects. Third, it yields a direct interpretation of reduced density matrix elements in terms of $2d$52 angular-momentum coupling. Fourth, it enables decoherence-free subsystems for collective $2d$53 noise, because in the UGA basis $2d$54 acts only on the $2d$55 register while leaving $2d$56 invariant (Burkat et al., 10 Jun 2025).

Two conceptual clarifications are recurrent. One is terminological: the transform is often described as unitary, but in the CAS presentation it is stated to be “strictly, an isometry on the CAS sector.” The other concerns implementability: the explicit algorithm is a circuit-level isometry between basis descriptions, not a factorized fermionic operator identity, and no simple factorization into one- and two-fermion operators is given. A further implication of the CAS results is that the usefulness of the QPT is not limited to symmetry resolution alone; it also provides the coordinate system in which CAS amplitudes become exactly MPS-compressible with polynomial bond dimension. The open challenges named in the QPT paper are extending the method to full spin-dependent Hamiltonians with $2d$57, exploiting higher symmetries such as point-group and spatial symmetries, and integrating the transform with error-mitigation and variational algorithms (Jnane, 17 Jun 2026, Burkat et al., 10 Jun 2025).