Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms with applications in the resolution of partial differential equations, Hamiltonian simulations, the loading of classical data on quantum computers (quantum state preparation) and many others. In this paper, we introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits. The depth, i.e., the number of layers of quantum gates of the quantum circuit, is reducible with respect either to the width, i.e, the number of ancilla qubits, or to the accuracy between the implemented operator and the target one. While exact methods have an optimal exponential scaling either in terms of size, i.e., the total number of primitive quantum gates, or width, approximate methods prove to be efficient for the class of diagonal operators depending on smooth, at least differentiable, functions. Our approach is general enough to allow any method for diagonal operators to become adjustable-depth or approximate, decreasing the depth of the circuit by increasing its width or its approximation level. This feature offers flexibility and can match with the hardware limitations in coherence time or cumulative gate error. We illustrate these methods by performing quantum state preparation and non-unitary-real-space simulation of the diffusion equation. This simulation paves the way to efficient implementations of stochastic models useful in physics, chemistry, biology, image processing and finance.Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms with applications in the resolution of partial differential equations, Hamiltonian simulations, the loading of classical data on quantum computers (quantum state preparation) and many others. In this paper, we introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits. The depth, i.e., the number of layers of quantum gates of the quantum circuit, is reducible with respect either to the width, i.e, the number of ancilla qubits, or to the accuracy between the implemented operator and the target one. While exact methods have an optimal exponential scaling either in terms of size, i.e., the total number of primitive quantum gates, or width, approximate methods prove to be efficient for the class of diagonal operators depending on smooth, at least differentiable, functions. Our approach is general enough to allow any method for diagonal operators to become adjustable-depth or approximate, decreasing the depth of the circuit by increasing its width or its approximation level. This feature offers flexibility and can match with the hardware limitations in coherence time or cumulative gate error. We illustrate these methods by performing quantum state preparation and non-unitary-real-space simulation of the diffusion equation. This simulation paves the way to efficient implementations of stochastic models useful in physics, chemistry, biology, image processing and finance.

arXiv:2404.02819Apr 3, 20240 citations

Efficient quantum circuits for diagonal unitaries without ancillas

Jonathan Welch, Daniel Greenbaum, Sarah Mostame +1 more

The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary in the basis, to that of finding the minimal-length Walsh-series approximation to the function f ( x ). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth.The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary in the basis, to that of finding the minimal-length Walsh-series approximation to the function f ( x ). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth.

arXiv:1306.3991New Journal of PhysicsMar 31, 2014115 citations

On the Synthesis of Quantum Circuits for Diagonal Operators in Quantum Computation

Jonathan Welch

Diagonal unitary operators are commonly found in many quantum algorithms. They find application as analytical potential operators for quantum simulation, as well as for complex oracles used in quantum searches. However, in order to implement a quantum algorithm on a given quantum device, each operator must be decomposed into a sequence of fault-tolerant, device-level instructions. In general, to implement an $n$ -qubit diagonal unitary {\\em exactly} on a quantum computer generally requires $2^{n+1}-3$ one- and two-qubit gates. However, for most practical implementations of diagonal unitaries, some degree of approximation will be necessary if the circuit is to be efficient. In this thesis we develop two complementary methods for the approximate synthesis of quantum circuits for diagonal unitaries. We show how to apply these techniques to real-space quantum simulation and show how efficient high fidelity quantum simulations can be implemented with low-depth quantum circuits.Diagonal unitary operators are commonly found in many quantum algorithms. They find application as analytical potential operators for quantum simulation, as well as for complex oracles used in quantum searches. However, in order to implement a quantum algorithm on a given quantum device, each operator must be decomposed into a sequence of fault-tolerant, device-level instructions. In general, to implement an $n$-qubit diagonal unitary {\\em exactly} on a quantum computer generally requires $2^{n+1}-3$ one- and two-qubit gates. However, for most practical implementations of diagonal unitaries, some degree of approximation will be necessary if the circuit is to be efficient. In this thesis we develop two complementary methods for the approximate synthesis of quantum circuits for diagonal unitaries. We show how to apply these techniques to real-space quantum simulation and show how efficient high fidelity quantum simulations can be implemented with low-depth quantum circuits.

Sep 29, 20152 citations

The power of constant-depth quantum circuits of unbounded size

Sergii Strelchuk, Sathyawageeswar Subramanian, Máté Weisz

Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth. We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential. The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy. Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension $d$ and $M\geq d^2-1$ ports, we construct a unitary circuit of depth $O(\sqrt d)$ , independent of $M$ and including resource preparation and port selection, with entanglement fidelity at least $(1-(d^2-1)/(2M))^2$ . Thus, at fixed $d$ , the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on $d$ can also be removed remains open.Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth. We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential. The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy. Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension $d$ and $M\geq d^2-1$ ports, we construct a unitary circuit of depth $O(\sqrt d)$, independent of $M$ and including resource preparation and port selection, with entanglement fidelity at least $(1-(d^2-1)/(2M))^2$. Thus, at fixed $d$, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on $d$ can also be removed remains open.

quant-pharXiv:2609.40294Oct 1, 20260 citations

Efficient quantum circuits for diagonal unitaries without ancillas

Jonathan Welch, Greenbaum, Daniel, Sarah Mostame +1 more

The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we point out a correspondence between Walsh functions and basis for diagonal operators that gives a simple way to construct efficient circuits for diagonal unitaries without ancillas. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary eif(ˆx) in the |xi basis, to that of finding the minimal-length Walsh-series approximation to the function f(x). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth.The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we point out a correspondence between Walsh functions and basis for diagonal operators that gives a simple way to construct efficient circuits for diagonal unitaries without ancillas. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary eif(ˆx) in the |xi basis, to that of finding the minimal-length Walsh-series approximation to the function f(x). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth.

Jan 1, 201435 citations

Operator learning for efficient quantum computation

Paul Over, Sergio Bengoechea, Leonardo Borello Busilacchi +3 more

An efficient implementation of quantum algorithms is often hindered by the lack of efficient primitives for operators and state preparation. This limits both the ability of near-term quantum hardware to simulate complex problems and the potential of fault-tolerant algorithms to achieve practical quantum advantage. To address this, we propose a framework that maps discrete operators onto compact, structured quantum circuits. The resulting variational circuits can be tailored to the connectivity and long-range interaction of the target hardware. The learning process employs backpropagation together with a cost function that efficiently optimizes unitary operators and non-unitary --dense or sparse-- operators using only a single ancilla qubit for block encoding. Additionally, we introduce a regularization term that reduces the approximation error. The approach is validated for both quantum mechanical and engineering applications. In the former case, we learn propagators that arise in native quantum problems --such as quantum simulation and quantum chemistry-- and achieve improved resource scaling in comparison to standard Suzuki--Trotter expansions. In the latter case, we demonstrate the approach's ability to implement the second-order central finite difference approximation of the Laplace operator --relevant for solving partial differential equations-- while improving upon current error metrics. The final example deals with learning a dense, non-unitary operator that arises in the analysis of inviscid potential flow around an airfoil. These results motivate the extension of the framework to a broader class of operator-synthesis problems beyond the prototypical engineering and quantum applications considered here.An efficient implementation of quantum algorithms is often hindered by the lack of efficient primitives for operators and state preparation. This limits both the ability of near-term quantum hardware to simulate complex problems and the potential of fault-tolerant algorithms to achieve practical quantum advantage. To address this, we propose a framework that maps discrete operators onto compact, structured quantum circuits. The resulting variational circuits can be tailored to the connectivity and long-range interaction of the target hardware. The learning process employs backpropagation together with a cost function that efficiently optimizes unitary operators and non-unitary --dense or sparse-- operators using only a single ancilla qubit for block encoding. Additionally, we introduce a regularization term that reduces the approximation error. The approach is validated for both quantum mechanical and engineering applications. In the former case, we learn propagators that arise in native quantum problems --such as quantum simulation and quantum chemistry-- and achieve improved resource scaling in comparison to standard Suzuki--Trotter expansions. In the latter case, we demonstrate the approach's ability to implement the second-order central finite difference approximation of the Laplace operator --relevant for solving partial differential equations-- while improving upon current error metrics. The final example deals with learning a dense, non-unitary operator that arises in the analysis of inviscid potential flow around an airfoil. These results motivate the extension of the framework to a broader class of operator-synthesis problems beyond the prototypical engineering and quantum applications considered here.

quant-pharXiv:2606.20184Quantum Science and TechnologyJun 19, 20260 citations

n

-qubit diagonal unitary {\\em exactly} on a quantum computer generally requires

2^{n+1}-3

one- and two-qubit gates. However, for most practical implementations of diagonal unitaries, some degree of approximation will be necessary if the circuit is to be efficient. In this thesis we develop two complementary methods for the approximate synthesis of quantum circuits for diagonal unitaries. We show how to apply these techniques to real-space quantum simulation and show how efficient high fidelity quantum simulations can be implemented with low-depth quantum circuits.

d

and

M\geq d^2-1

ports, we construct a unitary circuit of depth

O(\sqrt d)

, independent of

M

and including resource preparation and port selection, with entanglement fidelity at least

(1-(d^2-1)/(2M))^2

. Thus, at fixed

d

, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on

d

can also be removed remains open.