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arXiv:2606.12096

Necessary and Sufficient Conditions for Universal Gates with Pauli Strings and Beyond

Isaac D. Smith, Hans J. Briegel, Hendrik Poulsen Nautrup

quant-phSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate $\mathfrak{su}(2^n)$, i.e., to be universal for quantum computation on $n$ qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.

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Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate su(2n)\mathfrak{su}(2^n)su(2n), i.e., to be universal for quantum computation on nnn qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators SSS and nnn-qubit ``magic" Hamiltonians HHH that together generate the full Lie algebra su(2n)\mathfrak{su}(2^n)su(2n), with a special focus on the Ising Hamiltonian JJJ. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian HHH. We find that the number of applications of HHH needed to approximate any unitary up to accuracy εεε scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.
Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. When this set is taken to consist of only products of Pauli operators, we show that the minimal such set generating su(2^{N}) contains 2N+1 elements. We provide a number of examples of such generating sets and furthermore provide an algorithm for producing a sequence of rotations corresponding to any given Pauli rotation, which is shown to have optimal complexity. We also observe that certain sets generate su(2^{N}) at a faster rate than others, and we show how this rate can be optimized by tuning the fraction of anticommuting pairs of generators. Finally, we briefly comment on implications for measurement-based and trapped ion quantum computation as well as the construction of fault-tolerant gate sets.
In this paper, we design quantum circuits for the exponential of scalednnn-qubit Pauli-strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity andXXXgates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with thennn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Paulistring is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1n+1n+1)-qubit systems can be constructed fromnnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 2-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers, and observe that the noisy numerical simulation closely resembles the noiseless one when gate and idle errors are on the order ofO(10−3)O\left(10^{-3}\right)O(10−3)or smaller.
In this paper, we design quantum circuits for the exponential of scaled nnn-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and XXX gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the nnn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Pauli-string is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1)(n+1)(n+1)-qubit systems can be constructed from nnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 222-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers and observe that the noisy simulation closely resembles the noiseless one when gate and idle errors are on the order of O(10−3)O(10^{-3})O(10−3) or smaller.
This article deals with quantum computations in the Pauli basis whose elements are usually identified with the Pauli strings. This approach allows us to represent quantum states, observables, and unitary operators in the unified form of linear combination of Pauli strings, so that all operations can be reduced to the string compositions. Nevertheless a formal justification of the Pauli basis for quantum computations should be based on the strong results of complex linear algebra and the theory of Hilbert spaces. We briefly review the main features of Pauli strings for quantum states and unitary operators, and also the key operations with them, including an algorithm for string compositions and a transformation algorithm from the standard basis to the Pauli basis.
An essential element of classical computation is the ``if-then'' construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two-dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.