Representation of Boolean functions in terms of quantum computation

Abstract: The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic (k-valued logic, where k is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware.

• The paper develops a unified representation of Boolean and prime-valued logical functions through reversible quantum transformations using Zhegalkin polynomials and controlled-NOT-type circuit constructions.
• The method introduces a reversible embedding of classical functions $f$, where an input register $|x angle$ is preserved and the function value is accumulated in an output register $|yoplus f(x) angle$, providing an interface between truth tables and quantum circuits.
• The algebraic normal form of a Boolean function extends into polynomial representations that yield scalable circuit constructions, for single and multiple outputs, across prime-coded domains, leading to compact circuit configurations for reversible functions.

The paper develops a unified representation of Boolean and prime-valued logical functions through reversible quantum transformations, Zhegalkin polynomials, and controlled-NOT-type circuit constructions. Its central premise is that a classical function $f$ should not be represented as the generally noninvertible map $|x\rangle \mapsto |f(x)\rangle$, but through the reversible embedding

$|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$

where the input register is preserved and the function value is accumulated into an output register. This construction yields a unitary operator and provides the formal interface between truth tables, algebraic normal forms, and quantum circuits (1906.06374).

Reversible representation of Boolean functions

For an $n$-bit input and one-bit output, the unitary matrix associated with $f$ is block diagonal with $2^n$ blocks. The block corresponding to an input string $x$ is the identity matrix $I$ if $f(x)=0$ and the Pauli-$X$ matrix if $|x\rangle \mapsto |f(x)\rangle$0. Consequently, the matrix dimension is $|x\rangle \mapsto |f(x)\rangle$1, while the number of distinct functions is $|x\rangle \mapsto |f(x)\rangle$2.

This correspondence is elementary but structurally useful. For one input bit, the four Boolean functions are represented by the identity, controlled-NOT, an $|x\rangle \mapsto |f(x)\rangle$3 gate combined with controlled-NOT, and an unconditional $|x\rangle \mapsto |f(x)\rangle$4 gate. The block-matrix construction makes the reversible character explicit: function evaluation changes the target register while retaining sufficient information in the source register to make the transformation invertible.

The paper correctly emphasizes that the reversible embedding is more general than the conventional classical notation $|x\rangle \mapsto |f(x)\rangle$5. The latter discards the input and is not generally injective, whereas the transformation $|x\rangle \mapsto |f(x)\rangle$6 is a permutation of computational-basis states and therefore unitary. The construction is consequently suitable for quantum-oracle implementations, although the paper does not analyze oracle-query complexity, ancilla requirements, or gate synthesis cost beyond the direct circuit correspondence.

A minor typographical problem occurs in the statement that the number of block-diagonal matrices is $|x\rangle \mapsto |f(x)\rangle$7; the intended expression is evidently $|x\rangle \mapsto |f(x)\rangle$8. The subsequent examples use the correct values: $|x\rangle \mapsto |f(x)\rangle$9 functions for $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$0, $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$1 for $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$2, and $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$3 for $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$4.

Zhegalkin polynomials and circuit synthesis

The principal technical contribution is the use of tensor-product basis vectors to derive the Zhegalkin, or algebraic normal form, directly from a truth table. Over $|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$5, every Boolean function has a unique expansion

$|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$6

The paper constructs basis functions indexed by computational-basis strings. For two variables, the indicator functions are

$|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$7

$|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$8

These basis functions are mutually exclusive on computational-basis inputs and form a basis for the vector space of Boolean functions. Each basis function is associated with a coefficient vector obtained through tensor products of the one-variable vectors

$|x,y\rangle \mapsto |x,y\oplus f(x)\rangle,$9

The truth-table vector is decomposed as a mod-$n$0 sum of those basis functions whose corresponding truth-table entries are one. The same decomposition can be performed on their coefficient vectors. In this way, the method transforms a truth table into the coefficient vector of the Zhegalkin polynomial using tensor products and binary addition rather than repeated symbolic interpolation.

The resulting circuit synthesis rule is direct. The constant monomial $n$1 gives an $n$2 gate on the target. A linear monomial $n$3 gives a CNOT controlled by input qubit $n$4. A monomial involving $n$5 variables,

$n$6

gives an $n$7-controlled NOT gate targeting the output qubit. Since there are $n$8 possible monomials, the complete unoptimized basis contains $n$9 possible controlled operations, and selecting an arbitrary subset produces the $f$0 Boolean functions.

This establishes a one-to-one correspondence between algebraic-normal-form terms and a restricted family of reversible circuits. The correspondence is conceptually clean, but it should not be interpreted as an efficient synthesis result in the complexity-theoretic sense. A generic function may contain exponentially many nonzero Zhegalkin coefficients, and the direct circuit can require high-control gates that themselves must be decomposed into a native gate set. The paper provides a representation and construction procedure, not an optimized compilation algorithm.

Multiple-output functions

The framework is extended to functions with an $f$1-bit output,

$f$2

through the reversible map

$f$3

For each input $f$4, the output string $f$5 selects the block

$f$6

where $f$7 and $f$8. The full unitary therefore has dimension $f$9.

The half-adder illustrates the construction. Its two output bits are

$2^n$0

The sum output is generated by two CNOT gates, while the carry output is generated by a Toffoli gate. The corresponding reversible operator acts on four qubits and has dimension $2^n$1. The three-input adder is expressed as

$2^n$2

$2^n$3

and acts on five qubits, producing a $2^n$4 unitary matrix.

These examples demonstrate that the method handles vector-valued functions componentwise: each output column of the truth table is converted into its own algebraic normal form and then implemented on the corresponding target qubit. The paper does not address output-register initialization, garbage management, ancilla uncomputation, or the distinction between in-place arithmetic and general reversible embeddings. Those issues are essential for practical arithmetic circuit synthesis but are outside the presented formal construction.

Extension to prime-valued logic

A substantial part of the paper generalizes the binary method to $2^n$5-valued logic, with particular attention to prime $2^n$6. For three-valued logic, the binary shifts $2^n$7 and $2^n$8 are replaced by cyclic shift operators

$2^n$9

They satisfy

$x$0

A ternary truth-table value $x$1, $x$2, or $x$3 consequently selects $x$4, $x$5, or $x$6 as the corresponding block of the unitary transformation. This is the direct analogue of the binary $x$7 construction.

The polynomial representation is built from the evaluation vectors

$x$8

over $x$9. These vectors form a matrix $I$0, and the inverse matrix $I$1 supplies the coefficient vectors of the indicator functions. For three-valued logic,

$I$2

For an $I$3-argument function, the transformation is obtained by tensor products of these matrices. If $I$4 is the truth-table column and $I$5 is the coefficient column, then

$I$6

All arithmetic is performed modulo $I$7. The appendix verifies this procedure on a two-variable ternary example and obtains the coefficient vector through both explicit indicator-polynomial expansion and tensor-product matrix multiplication.

The same construction is illustrated for five-valued logic. For a one-variable function with values $I$8, multiplication by the displayed $I$9 matrix gives the coefficient vector

$f(x)=0$0

For a two-variable function over $f(x)=0$1 with $f(x)=0$2 truth-table entries, the coefficient vector has length $f(x)=0$3. The paper reports an explicit example with only two nonzero truth-table values and gives the resulting coefficient vector. These examples support the algebraic procedure, but they do not constitute performance benchmarks: no runtime, memory, circuit-depth, or gate-count comparisons are reported.

The restriction to prime $f(x)=0$4 is mathematically important. The construction relies on inversion of a Vandermonde-like evaluation matrix over a field. For prime $f(x)=0$5, the integers modulo $f(x)=0$6 form $f(x)=0$7, ensuring that every nonzero element is invertible. The paper states that composite values require a different treatment and that the presented completeness argument applies only to prime $f(x)=0$8. This qualification limits the claimed generality of the phrase “$f(x)=0$9-valued logic”: the fully established algebraic framework is for prime cardinalities, while the composite case is only described by analogy.

Quantum interpretation of ternary shifts

The ternary shift operators are also expressed as exponentials of a Hermitian generator. The paper gives a Hermitian matrix $X$0 such that a rotation by $X$1 produces $X$2, while a rotation by $X$3 produces $X$4. The eigenvalues of $X$5 are identified with the spin-1 projections $X$6, with logical labels obtained by the shift $X$7.

This section connects discrete cyclic permutations with continuous unitary evolution. For intermediate rotation angles, the operator need not preserve the computational basis. It instead creates superpositions. The paper gives the example that a $X$8 transformation applied to $X$9 produces

$|x\rangle \mapsto |f(x)\rangle$00

The amplitudes are consistent with normalization because the squared magnitudes sum to one. The example shows that the discrete logical shifts are embedded in a larger continuous unitary family. However, the physical interpretation in terms of spin-1 systems should be distinguished from the abstract qutrit construction: the existence of a three-dimensional Hilbert space does not by itself require a physical spin-1 implementation, and the paper does not specify a hardware Hamiltonian or experimental realization.

Quantitative scope and claims

The paper makes several combinatorial claims that clarify the scale of the representation:

• A one-bit Boolean function has $|x\rangle \mapsto |f(x)\rangle$01 possible truth tables.
• A two-bit Boolean function has $|x\rangle \mapsto |f(x)\rangle$02 possible truth tables.
• A three-bit Boolean function has $|x\rangle \mapsto |f(x)\rangle$03 possible truth tables.
• A three-valued one-input function has $|x\rangle \mapsto |f(x)\rangle$04 possibilities.
• A three-valued two-input function has $|x\rangle \mapsto |f(x)\rangle$05 possibilities.
• A three-valued three-input function has approximately $|x\rangle \mapsto |f(x)\rangle$06 possibilities.

The general count for $|x\rangle \mapsto |f(x)\rangle$07 inputs and $|x\rangle \mapsto |f(x)\rangle$08 possible output values is $|x\rangle \mapsto |f(x)\rangle$09, not a specifically quantum phenomenon. The quantum contribution is the reversible realization of each truth table as a controlled shift operator. Similarly, the number of Boolean circuits in the paper’s unrestricted monomial-selection construction equals the number of Boolean functions because each monomial is either included or omitted. This equality demonstrates representational completeness, but it does not imply that all functions have equally efficient circuits or that the proposed circuits are minimal.

The paper’s strongest claim is that the method can “significantly improve” the transition from classical logic to quantum hardware. The constructions substantiate a systematic translation from truth tables and algebraic normal forms to reversible circuits. They do not, however, provide evidence for an improvement in hardware metrics. No experimental implementation, synthesis benchmark, asymptotic gate-complexity analysis, noise analysis, or comparison with established reversible-circuit synthesis methods is given. The claim is therefore supported as a conceptual and algorithmic framework, but not as a demonstrated hardware-performance result.

Limitations and open questions

The main limitation is the absence of complexity analysis. Tensor-product construction is algebraically simple, but the vectors and matrices have dimensions that grow as $|x\rangle \mapsto |f(x)\rangle$10. For binary functions, the truth table itself already has $|x\rangle \mapsto |f(x)\rangle$11 entries, and the coefficient vector has the same dimension. Explicit tensor-product matrices can therefore become infeasible well before the corresponding Boolean function is physically synthesized. The paper does not establish whether its method can be implemented implicitly, sparsely, or through fast transforms.

The circuit correspondence also assumes access to multi-controlled NOT operations, or their $|x\rangle \mapsto |f(x)\rangle$12-valued analogues. In a realistic gate library, these operations require decomposition, often using ancillas and additional depth. Consequently, the number of polynomial terms is not a complete measure of implementation cost. The paper leaves open whether the polynomial representation can be combined with minimization, Reed–Muller optimization, ancilla reduction, or hardware-aware compilation.

For multiple-output functions, each output is treated independently, but the paper does not discuss shared subexpressions between output polynomials. Such sharing could substantially reduce gate counts for arithmetic functions. It also does not analyze whether the resulting transformations preserve clean ancilla states or how the circuits should be inverted and composed within larger quantum algorithms.

Finally, the prime-dimensional generalization is algebraically well motivated but physically underdeveloped. The shift operators define valid unitary qutrit or qudit transformations, yet the paper does not identify a universal native gate set, decomposition theorem, or implementation model for the resulting controlled shifts. The relationship between the polynomial basis over $|x\rangle \mapsto |f(x)\rangle$13 and experimentally relevant qudit architectures remains an open question within the paper’s framework.

Conclusion

The paper presents a coherent algebraic route from truth tables to reversible quantum transformations. Its key results are the block-diagonal representation of Boolean functions, the tensor-product method for computing Zhegalkin-polynomial coefficients, and the term-by-term mapping from polynomial monomials to controlled quantum gates (1906.06374). The extension to multiple outputs and prime-valued logic broadens the framework beyond binary single-output functions.

The contribution is primarily structural rather than performance-oriented. It establishes representational completeness and gives explicit constructive procedures, including numerical examples for binary, ternary, and five-valued logic. The unresolved issues concern scalability, gate decomposition, circuit optimization, ancilla management, and physical realization of higher-dimensional controlled shifts.