Diagonal Walsh Operators

• Diagonal Walsh operators are a set of commuting, diagonal operators defined via Walsh functions that form an orthonormal basis for n-qubit diagonal matrices.
• They enable resource-efficient synthesis and precise encoding of functions like potential energy surfaces and CI wavefunctions without requiring ancilla qubits.
• Their application in quantum simulation and variational quantum configuration interaction provides scalable circuit depth and robust error control even for non-unitary operations.

Diagonal Walsh operators are a class of commuting, diagonal operators on qubit registers whose structure, basis, and circuit application are built from the theory of Walsh functions. They provide an orthonormal operator basis for the construction and efficient circuit synthesis of arbitrary diagonal unitaries and non-unitary diagonal operators on $n$-qubit (or $r$-qubit) Hilbert spaces, with direct implications for quantum simulation, variational quantum eigensolvers, and quantum chemistry. The construction of these operators and their resource-efficient compilation enable exact or approximate encoding of functions—such as potential energy surfaces or configuration interaction (CI) wavefunctions—without reliance on ancilla qubits and with favorable scaling in gate count and circuit depth (Welch et al., 2013, Aydoğan et al., 11 Jan 2026).

1. Mathematical Foundation: Walsh Functions and Operator Basis

Walsh functions $w_k(x)$ are defined on the integers $x \in \{0,\ldots,2^n-1\}$, written in binary as $x = (x_1,\ldots,x_n)$, $x_i \in \{0,1\}$, while $k$ has binary digits $(k_1,\ldots,k_n)$. The Paley-ordered Walsh functions are

$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$

or, equivalently, with inner product modulo 2,

$$w_j(k) = (-1)^{\langle j,k \rangle}, \quad \langle j,k \rangle = \sum_{i=1}^{r} j_i k_i \bmod 2$$

where $r$0 index the basis elements.

The set $r$1 forms an orthonormal basis for all real-valued functions on bitstrings of length $r$2, satisfying

$r$3

Promoted to operators on the $r$4-qubit Hilbert space, the diagonal Walsh operator is

$r$5

where $r$6 is the Pauli-$r$7 acting on the $r$8-th qubit. There are $r$9 such operators, each diagonal in the computational basis, with eigenvalues $w_k(x)$0. The set $w_k(x)$1 provides an orthonormal basis (under the Hilbert–Schmidt product) for all $w_k(x)$2 diagonal matrices (Welch et al., 2013, Aydoğan et al., 11 Jan 2026).

2. Expansion and Synthesis of Diagonal Operators

Any diagonal operator $w_k(x)$3 can be expanded in the Walsh basis: $w_k(x)$4 For diagonal unitaries generated as $w_k(x)$5, with $w_k(x)$6 a real function on bitstrings, the unique Walsh expansion is

$w_k(x)$7

and the unitary decomposes as

$w_k(x)$8

Due to the commutativity of all $w_k(x)$9, the ordering is irrelevant.

In quantum chemistry (e.g., variational quantum configuration interaction), CI amplitudes $x \in \{0,\ldots,2^n-1\}$0 can be encoded via the Walsh–Fourier decomposition, mapping each real amplitude into phases $x \in \{0,\ldots,2^n-1\}$1 on Pauli–$x \in \{0,\ldots,2^n-1\}$2 strings (Aydoğan et al., 11 Jan 2026).

3. Resource-Efficient Circuit Realization

Each elementary diagonal Walsh operator $x \in \{0,\ldots,2^n-1\}$3 corresponds to a multi-controlled $x \in \{0,\ldots,2^n-1\}$4 rotation. The efficient decomposition proceeds by:

• Identifying the most significant nonzero bit $x \in \{0,\ldots,2^n-1\}$5 in $x \in \{0,\ldots,2^n-1\}$6.
• Applying $x \in \{0,\ldots,2^n-1\}$7 to qubit $x \in \{0,\ldots,2^n-1\}$8.
• For each other $x \in \{0,\ldots,2^n-1\}$9 with $x = (x_1,\ldots,x_n)$0, surrounding $x = (x_1,\ldots,x_n)$1 with a pair of CNOTs: control at $x = (x_1,\ldots,x_n)$2, target at $x = (x_1,\ldots,x_n)$3.

Optimizing the circuit, consecutive terms $x = (x_1,\ldots,x_n)$4 in a Gray (sequency) order (where consecutive $x = (x_1,\ldots,x_n)$5 differ by one bit) allow maximal CNOT cancellation, yielding a circuit of depth $x = (x_1,\ldots,x_n)$6 for $x = (x_1,\ldots,x_n)$7 terms (full expansion: $x = (x_1,\ldots,x_n)$8 gates) (Welch et al., 2013).

For non-unitary diagonal operators encountered in VQCI, the operator $x = (x_1,\ldots,x_n)$9 is embedded via a dilation into a larger unitary $x_i \in \{0,1\}$0 on $x_i \in \{0,1\}$1 qubits, implemented as a product over exponentials of commuting extended Walsh operators acting on both system and ancilla (Aydoğan et al., 11 Jan 2026).

4. Complexity, Scaling, and Sparse Approximations

Resource estimates are controlled by the number $x_i \in \{0,1\}$2 of nonzero Walsh coefficients used in the expansion:

• Full expansion: $x_i \in \{0,1\}$3 (all Walsh terms), gate-count $x_i \in \{0,1\}$4.
• Approximate expansion (smooth $x_i \in \{0,1\}$5): For $x_i \in \{0,1\}$6-accurate approximation, $x_i \in \{0,1\}$7 terms suffice, independent of $x_i \in \{0,1\}$8 for $x_i \in \{0,1\}$9.
• Sparse approximation: Retaining only largest $k$0 coefficients yields operator-norm error $k$1; gate count $k$2.
• CI encoding (minimal surjective set): Number of terms $k$3 for $k$4 selected determinants gives CNOT count $k$5; for partial expansion $k$6, CNOT count $k$7 (Aydoğan et al., 11 Jan 2026).

Ancilla-free realization is possible for unitaries; the block-ancilla construction appears only in the non-unitary (norm-not-preserving) VQCI context.

5. Applications in Quantum Simulation and VQCI

Diagonal Walsh operators provide highly efficient encoding mechanisms for major classes of quantum algorithms:

• Quantum simulation: Real-space simulation of Trotter steps $k$8, with sampled $k$9, maps directly onto sparse Walsh expansions. For the classical Eckart barrier problem, high-fidelity simulation ($(k_1,\ldots,k_n)$0) is achieved with circuit depth $(k_1,\ldots,k_n)$1 and only 30 largest Walsh components for $(k_1,\ldots,k_n)$2 qubits (Welch et al., 2013).

  n(qubits)	$(k_1,\ldots,k_n)$3 (error)	M (#terms)	Gate count	Simulation Fidelity
  10	exact	1023	2045	1.00
  8	~5%	30	60	~0.98
  7	~10%	19	38	~0.91
  6	~15%	14	28	~0.65

• VQCI for electronic structure: By an initial Dicke or quantum-walk state-preparation, followed by Walsh-encoded diagonal CI amplitude block, the entire configuration interaction (CI) space is compiled with gate counts $(k_1,\ldots,k_n)$4–$(k_1,\ldots,k_n)$5 (for $(k_1,\ldots,k_n)$6 determinants), enabling practical scaling to large systems. Fidelity benchmarks for H$(k_1,\ldots,k_n)$7, H$(k_1,\ldots,k_n)$8, H$(k_1,\ldots,k_n)$9O, LiH, BeH$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$0, NH$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$1 confirm chemical accuracy and outperform coupled-cluster in strongly correlated regimes (Aydoğan et al., 11 Jan 2026).

  Molecule	Qubits	Determinants $w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$2	Subspace Prep CNOTs	Walsh Ansatz CNOTs	Total
  H$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$3	12	400	52	726	778
  H$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$4	16	$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$5	391	8756	9147
  LiH	12	400	96	520	616
  NH$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$6	16	$w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$7	218	7530	7748

This construction precludes barren-plateau trainability issues: the commuting nature of Walsh operators ensures the absence of 2-design structure in the ansatz.

6. Contextual Significance and Limitations

The resource scaling and modularity of diagonal Walsh operator schemes address a major barrier in quantum algorithm design—efficient implementation of diagonal operations, historically resource-bottlenecked and often requiring many ancilla qubits. The approach's absence of overparameterization and resilience to trainability plateaus (as seen in unitary coupled-cluster circuits) are central to its adoption in contemporary quantum algorithms in chemistry and other fields (Aydoğan et al., 11 Jan 2026).

A plausible implication is that as quantum devices scale beyond $w_k(x) = (-1)^{\sum_{i=1}^n k_i x_i}$8 qubits and as determinant subspace selection for CI becomes dominant in practical simulations, the advantages of Walsh operator sparsity and circuit efficiency could be increasingly central in quantum-classical hybrid methodologies.

7. References and Research Trajectory

The foundational construction and ancilla-free circuit compilation of diagonal unitaries using the Walsh basis were established in Welch, Greenbaum, Mostame, and Aspuru-Guzik (Welch et al., 2013), and systematically extended to non-unitary, full and partial CI encoding in variational quantum chemistry (VQCI) by Aydoğan, Gross, and coauthors (Aydoğan et al., 11 Jan 2026). As quantum algorithms for simulation and eigensolving evolve, the diagonal Walsh operator formalism continues to underpin efficient, scalable approaches to problems where diagonal structure and classical decomposability coincide with quantum resource constraints.

Key References:

• Welch, Greenbaum, Mostame, Aspuru-Guzik, "Efficient Quantum Circuits for Diagonal Unitaries Without Ancillas" (Welch et al., 2013)

• Aydoğan, Gross et al., "Subspace Selected Variational Quantum Configuration Interaction with a Partial Walsh Series" (Aydoğan et al., 11 Jan 2026)