Hamiltonian engineering of general two-body spin-1/2 interactions

Spin Hamiltonian engineering in solid-state systems plays a key role in a variety of applications ranging from quantum information processing and quantum simulations to novel studies of manybody physics. By analyzing the irreducible form of a general two-body spin-1/2 Hamiltonian, we identify all interchangeable interaction terms using rotation pulses. Based on this, we derive novel pulse sequences, defined by an icosahedral symmetry group, providing the most general achievable manipulation of interaction terms. We demonstrate that, compared to conventional Clifford rotations, these sequences offer advantages for creating Zeeman terms essential for magnetic sensing, and could be utilized to generate new interaction forms. The exact series of pulses required to generate desired interaction terms can be determined from a linear programming algorithm. For realizing the sequences, we propose two experimental approaches, involving pulse product decomposition, and off-resonant driving. Resulting engineered Hamiltonians could contribute to the understanding of many-body physics, and result in the creation of novel quantum simulators and the generation of highly-entangled states, thereby opening avenues in quantum sensing and information processing.

Spin Hamiltonian engineering in solid-state systems plays a key role in a variety of applications ranging from quantum information processing and quantum simulations to novel studies of manybody physics. By analyzing the irreducible form of a general two-body spin-1/2 Hamiltonian, we identify all interchangeable interaction terms using rotation pulses. Based on this, we derive novel pulse sequences, defined by an icosahedral symmetry group, providing the most general achievable manipulation of interaction terms. We demonstrate that, compared to conventional Clifford rotations, these sequences offer advantages for creating Zeeman terms essential for magnetic sensing, and could be utilized to generate new interaction forms. The exact series of pulses required to generate desired interaction terms can be determined from a linear programming algorithm. For realizing the sequences, we propose two experimental approaches, involving pulse product decomposition, and off-resonant driving. Resulting engineered Hamiltonians could contribute to the understanding of many-body physics, and result in the creation of novel quantum simulators and the generation of highly-entangled states, thereby opening avenues in quantum sensing and information processing.
For many decades, rotation pulse sequences have been utilized in nuclear magnetic resonance (NMR) for manipulating spin states through Hamiltonian engineering [2,3]. In recent years, solid-state spin systems such as defects in diamonds, silicon, and silicon carbide have emerged as useful platforms for quantum technologies, thereby reviving the neccesity in efficient spin control schemes. In particular, the Nitrogen-Vacancy (NV) centers in diamond, which offer optical initialization and readout capabilities, and can be treated to some extent as spin-1/2 qubits, are widely used for sensing [4][5][6][7][8][9][10] and quantum information processing [11][12][13][14]. Manipulating the dipolar interactions within an ensemble of such spins could pave the way towards novel studies of manybody dynamics [15][16][17], the creation of quantum simulators and sensors [18], and generation of non-classical spin states [17,19]. Recent studies of such Hamiltonian engineering, analyzing the effects of control pulses from the Clifford rotation group, resulted in a novel scheme of generating certain types of Hamiltonians [18].
Here, we use group theory to go beyond previous work, introducing a more general platform of interaction manipulations, namely pulse sequences defined by an icosahedral symmetry. We emphasize that such pulses can provide the most general achievable manipulations of interaction terms. In particular, by utilizing a linear programming algorithm, we derive the proper sequences for transforming the natural NV-NV dipolar interaction Hamiltonian to several target Hamiltonians providing novel applications, which could not be generated using conventional Clifford rotations.
We begin by introducing the general interaction Hamiltonian of N spin-1/2 particles, which contains up to two-body interaction terms, where σ represents the Pauli spin operators, a, b ∈ {1 . . . N } the spin indices in the ensemble, T ( n) is the matrix (vector) of coefficients for the two-body interaction (one-spin) terms, and i, j ∈ {x, y, z}. The coefficient matrix can be written in the irreducible form with and γ (ab) is extracted from the traceless symmetric matrix  where PS (1) and PS (2) are 3 × 3 and 5 × 5 matrices uniquely representing the effects of the pulse sequence on the interaction terms [21]. These matrices correspond to convex combinations of the three dimensional matrix representing rotation around the axisn by a general angle 0 < θ < 2π, R 3×3 n,θ , and of a 5 × 5 reduced form of R 3×3 n,θ ⊗ R 3×3 n,θ respectively [21] (after removing the redundancy in the traceless symmetric matrix). Here, we will focus on specific rotation angles θ and axesn characterizing two symmetry groups, the Clifford group and the regular icosahedral group, demonstrating that the latter provides the most general interaction manipulation capabilities. Note that rotation pulses cannot modify the isotropic interaction part σ (a) · σ (b) .
Forming the Clifford rotation group, one basic family of spin control sequences consists of a series of ( π 2 )rotation pulses {P k } applied around the three principle axes of the Bloch-sphere at times {t k } [18]. For adjacent pulses with zero spacings (t j − t j−1 = 0 for a given j), such a structure considers rotations by any integer multiplying angles of ( π 2 ). The novel utilization of such sequences, which were obtained by linear programming, was recently proposed for the engineering of the one-spin Hamiltonian proportional to a σ (a) z [18]. We extend this work by studying the general Hamiltonian engineering capabilities of the Clifford group, as well as a higher dimensional icosahedral symmetry group.
In considering the structure of matrices forming the Clifford rotation group [22], although the general structure of PS (1) can contain nine nonzero elements, the form of PS (2) must consist of two independent (2×2 and 3×3) blocks, with all other elements equal to zero. As a result, the capabilities of interchanging different interaction terms are limited. While the isotropic part σ (a) · σ (b) cannot be manipulated at all, different interaction terms inside the vector λ . For example, the initial interaction term σ (a) x by applying sequences consisting of solely ( π 2 )-pulses. The structure of matrices forming the regular icosahedral symmetry group [22], however, provides a different picture: the application of a series of 2π 5 -pulses, at angles determined by this symmetry group, imposes no limitations upon the elements of the matrix PS (2) . As a result, such icosahedral pulses enable interchanging different interaction terms inside the vector λ (ab) 2 , thereby providing the most general manipluation capabilities of two spin-interaction terms. Utilizing icosahedral pulses could transfer an interaction term of the form σ (a) x , which could not be generated solely by Clifford rotations.
Extracting the right pulse sequence that transforms an initial Hamiltonian to a desired one involves utilizing linear programming [18]. Given a matrix A, a vector b and lower and upper bounds l b , u b , such an algorithm is designed to find the vector x satisfying between the limits l b < x < u b , while minimizing the l 1 norm of the solution. The problem of identifying the right sequence for Hamiltonian engineering is mapped onto this algorithm in the following way [ (2)], we denote A = ( v i ⊗ 1)R, where R is a matrix whose different columns represent the effects of different rotations from the symmetry group [21]. This way, for a sequence with a total cycle time T , the k'th element of the solution vector to (4), x k = τ k T , will represent the duration within the sequence τ k associated with the rotation R k (for excluded rotations, x k = 0 ). As the actual applied control pulses inverting this relation will identify the desired pulse sequence. Additional symmetrization was applied to cancel higher (even) orders of average Hamiltonian terms. Moreover, initial and final pulses were introduced to satisfy the cyclic condition of the average Hamiltonian theory for a sequence with n pulses,H 1 =H n [2,3,20,21], and pulses with nontrivial angles and axes were decomposed (as composite pulses) to products of π 2 and 2π 5pulses.
One of the most promising platforms for studying many-body dynamics in the solid-state is the optically addressable spin ensemble of NV centers. We now discuss, within the framework of the average Hamiltonian theory, several useful target Hamiltonians that could be engineered by pulse sequences affecting the initial NV-NV interaction Hamiltonian (essentially relevant for general dipole coupling). Considering a two-level manifold (e.g. m s = 0, 1) of the NV ground states with a particular crystallographic orientation, in the rotating frame with respect to the bare Hamiltonian, and under the rotating wave approximation, such an interaction Hamiltonian is given by [15,17,23] where the dipolar interaction strengths ω (ab) between spins a and b depend on their relative positions, γ is the electron gyromagnetic ratio, and B z corresponds to an external magnetic field. The second term of the righthand side of (5) represents dipolar interactions, while the first term, often referred to as the "Zeeman term", is essential for magnetic sensing. First, we verify the validity of the linear programming procedures by reproducing the well-known WAHUHA sequence [2] [ Fig. 2 (a)], which was designed in NMR to decouple spin-1/2 dipolar interaction, while flipping the Zeeman term along the three axes of the Bloch-sphere. Due to the inequivalency between the two-level manifold of NV centers and spin-1/2 systems, the dipolar terms are not decoupled completely, and the isotropic part H iso = a<b ω (ab) σ (a) · σ (b) Subsequently, we utilize the linear programming algorithm for extracting the sequences required for generating several useful Hamiltonians (Figs. 2, 4). By considering Clifford rotations only, our code reproduces the HoRD-qubit-5 pulse sequence obtained by O'Keeffe et al. [18] [ Fig. 2 (b)], for decoupling dipolar interactions while keeping the one-spin Zeeman Hamiltonian, which is essential for magnetic sensing.
By extending spin control to pulses oriented along the angles of an icosahedron, we reveal a novel 14pulse sequence consisting of ( 2π 5 ) and ( 4π 5 )-pulses, which is expected to generate a similar Zeeman term with strength greater than HoRD-qubit-5 (∼ 1 2 versus 1 3 , as emphasized by simulations in Fig. 3). We now detail this derivation: starting from the initial NV Hamiltonian (5), a two-body interaction term with strength ω can be written according to Eq.  Fig. 2 (c)], which is greater than the 1 3 coefficient produced by the HoRD-qubit-5 sequence. In Fig. 3 we compare between simulated dynamics under the pure Zeeman Hamiltonian with the average Hamiltonians obtained using the HoRD sequence, the icosahedral sequence and its symmetrized variant. The clear advantage of the latter is evident. As a result, utilizing such icosahedral pulses could potentially result in enhanced magnetic sensitivities.
The conventional experimental realization of NV spin control utilizes in-phase-and-quadrature (IQ) modulated microwave (MW) signal, generating on-resonant pulses along the x − y plane of the Bloch-sphere. For realizing the icosahedral scheme, a pulse along the x − z or the y − z plane can be decomposed into products of three pulses along the x−y plane [21]. An alternative approach, avoiding the necessity for extra pulses and dealing with accumulating pulse imperfections, involves off-resonant pulses [21]. Spin dynamics simulations under Hamiltonians generated considering both realistic experimental approaches coincide with the theoretical application of ideal pulses [21].
The advantages of icosahedral pulse sequences can be further emphasized by applying a novel 10-pulse sequence [ Fig. 4(a)] on the NV ensemble with no external field (B z = 0), resulting in the interaction term σ (a) x σ (b) y , which cannot be generated solely by Clifford rotations.