Majorana States’ Phases Unlock Efficient Quantum Simulations

Scientists have detailed the phase-sensitive representation of Majorana stabilizer states, a crucial step towards advancing error correction and simulation techniques in information science. Tomislav Begušić and Garnet Kin-Lic Chan, from the California Institute of Technology, present a comprehensive analysis of these states and develop algorithms for calculating their amplitudes, inner products, and transformations under Majorana Clifford gates. This research is significant because it extends the understanding of stabilizer states to the realm of fermionic systems via the Majorana Clifford group, offering a pathway to efficiently represent and manipulate highly entangled states relevant to complex many-body physics simulations. The work establishes a foundation for exploring novel quantum algorithms and improving the robustness of quantum information processing.

New work details how to precisely capture the phase information within a specific type of quantum state known as a Majorana stabilizer state. This effort documents the phase-sensitive form of the corresponding Majorana stabilizer states, as well as the algorithms for computing their amplitudes, their inner products. They can be difficult to represent using other common strategies, such as computational basis expansion or tensor networks. In fermionic systems, and especially in quantum chemistry. The connection with concepts from quantum information science has typically proceeded by transforming into the qubit picture.

Here, this can lead to a loss of interpretability, given that typical transformations, such as the Jordan-Wigner transformation, map local fermionic operators into non-local Pauli operators. To use them as a basis for representing arbitrary fermionic quantum states, researchers need a phase-sensitive representation of Majorana stabilizer states.

Analogous to qubit-based stabilizer states, a phase-sensitive representation can be achieved either without phase information, sufficient for evaluating expectation values and sampling, or including the global phase, needed to represent general states as superpositions. These algorithms are sufficient to operate with fermionic states represented as superpositions of Majorana stabilizer states.

Scientists reserve c and c for denoting Majorana operators, a (a†) for the fermionic annihilation (creation) operators, and use a short form |0⟩= |000.0.0.0⟩to denote the vacuum state of a n-site fermionic system. For a binary vector x, they call a binary vector x its parity representation with definition xk = k X j=0xj, and a binary value π(x) = Pn−1 j=0 xj ∈{0, 1} the parity of x.

All addition of binary numbers is assumed to be mod 2 (logical exclusive OR operation or addition on Z2) — the number of nonzero elements of a binary array x is denoted by |x|. The notation where b has n elements is used. Single-particle Majorana operators are defined as cj and pj. Or a general Majorana operator Γ necessitates a more involved process, scaling at O(n2).

Meanwhile, this difference arises from the potential for these operations to affect all parameters defining the Majorana stabilizer state. Yet, the computation of probability amplitudes, ⟨x|ψ⟩, presents a different scaling behaviour. Its developed allow for this calculation in O((|x| + c)n) time. Where |x| represents the length of the computational basis state and c is a constant.

Here, this efficiency stems from reducing the exponential sum of Majorana operators to a single operator with a non-zero off-diagonal matrix element, simplifying the evaluation process. Once this reduction is achieved, the amplitude can be determined with a computational cost directly proportional to the length of the state vector and the system size, n. Still, determining the inner product between two Majorana stabilizer states, ⟨φ|ψ⟩, proves to be the most computationally demanding task.

In turn, the developed method achieves this in O(n3) time. Meanwhile, this cubic scaling originates from two primary factors: the construction of stabilizer tableaux for U†C, which involves matrix products. At the same time, the application of UC to a computational basis state. The significance extends beyond merely refining existing techniques, as this effort provides a framework for accurately tracking the amplitude and inner products of these states, essential for simulating quantum systems.

Since simulating fermionic systems is computationally demanding, these new algorithms offer a potential pathway to more efficient modelling of materials with exotic properties. Practical implementation remains a hurdle, as scaling these calculations to larger, more complex systems will require substantial computational resources. Still, the development of these algorithms could have a considerable impact on fields like condensed matter physics and materials discovery.

By providing a more efficient way to simulate fermions, scientists may be able to predict the behaviour of novel materials with greater accuracy — once these simulations become more accessible, the design of new superconductors or topological materials could be accelerated. Beyond this, The project opens avenues for exploring more advanced quantum error correction schemes, potentially leading to more stable and reliable quantum computers, and at present, the focus is on refining these algorithms and exploring their limitations. But the long-term implications for both fundamental research and technological advancement are considerable.