Whereas boson coherent states provide an elegant, intuitive and useful representation, we show that the desirable features of boson coherent states do not carry over very well to fermion fields unless one is prepared to use exotic approaches such as Grassmann fields. Specifically, we identify four appealing properties of boson coherent states (eigenstate of annihilation operator, displaced vacuum state, preservation of product states under linear coupling, and factorization of correlators) and show that fermion coherent states, and approximations to fermion coherent states, defined over the complex field, do not behave well for any of these four criteria.Whereas boson coherent states provide an elegant, intuitive and useful representation, we show that the desirable features of boson coherent states do not carry over very well to fermion fields unless one is prepared to use exotic approaches such as Grassmann fields. Specifically, we identify four appealing properties of boson coherent states (eigenstate of annihilation operator, displaced vacuum state, preservation of product states under linear coupling, and factorization of correlators) and show that fermion coherent states, and approximations to fermion coherent states, defined over the complex field, do not behave well for any of these four criteria.

Jul 18, 20050 citations

Exponential lower bounds on the fermionic Gaussian rank of magic states and the bosonic coherent state rank of Fock states

Oliver Reardon-Smith

Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of $\lvert M\rangle^{\otimes k}$ consists of $Ω(1.4^k)$ terms. Here $\lvert M\rangle$ is the most standard magic state for fermionic linear optics: the $4$ qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the $δ$ -approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of $1-δ^2$ . Our results on exact fermionic Gaussian rank apply directly to any product of $k$ fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an $n$ -mode bosonic Fock state with $m_j$ bosons in mode $j$ is exactly $\prod_{j}(1+m_j)$ , answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of $\lvert M\rangle^{\otimes k}$ consists of $Ω(1.4^k)$ terms. Here $\lvert M\rangle$ is the most standard magic state for fermionic linear optics: the $4$ qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the $δ$-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of $1-δ^2$. Our results on exact fermionic Gaussian rank apply directly to any product of $k$ fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an $n$-mode bosonic Fock state with $m_j$ bosons in mode $j$ is exactly $\prod_{j}(1+m_j)$, answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.

quant-pharXiv:2610.03699Oct 5, 20260 citations

Coherent States for Fermions

Chon‐Fai Kam, Wei-Min Zhang, D.H. Feng

Lecture notes in physicsNov 2, 20222 citations

Fermionic Coherent States

Monique Combescure, Didier Robert

Theoretical and mathematical physicsJan 1, 20122 citations

The lie algebraic structure of the set of one-particle fermion operators

Brian Weiner

The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).

International Journal of Quantum ChemistryMar 12, 19881 citation

\lvert M\rangle^{\otimes k}

consists of

Ω(1.4^k)

terms. Here

\lvert M\rangle

is the most standard magic state for fermionic linear optics: the

4

qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the

δ

-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of

1-δ^2

. Our results on exact fermionic Gaussian rank apply directly to any product of

k

fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an

n

-mode bosonic Fock state with

m_j

bosons in mode

j

is exactly

\prod_{j}(1+m_j)

, answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.