Computer scientists and others who are interested in learning more about quantum computation might be exposed, or re-exposed, to various concepts and classes of matrices from linear algebra. For example because of familiarity with truth-tables, (classical) reversible operators are often introduced in conjunction with unitary matrices.

Within quantum mechanics, there are a number of classes of matrices that are commonly used:

  1. As mentioned, unitary matrices are matrices $U$ such that $U^\dagger U = UU^\dagger = I$. These unitary matrices form the basis for discussion within the gate model of computation.
  2. Furthermore hermitian matrices are matrices $A$ such that $A = A^\dagger$. These matrices form the basis for, among other things, adiabatic computation.
  3. Additionally there are matrices, such as the creation $a^\dagger$, annihilation $a$, and number $N=a^\dagger a$ matrices, commonly referred to as ladder operators.

The familiar Pauli matrices $X$, $Y$, and $Z$ are both hermitian and unitary, while the creation and annihilation matrices are neither. Nonetheless reviewing Feynman's 1985 paper "Quantum Mechanical Computers", it appears that Feynman envisioned programming a quantum computer with an algebra or a calculus of sorts with sums and products of these ladder operators; indeed, he considered three qubits $a,b,c$ with $a$ and $b$ controlling the negation of $c$, and wrote a Toffoli gate explicitly as:

$$\mathsf{CCNOT}=1+a^\dagger ab^\dagger b(c+c^\dagger-1).$$

But other than Feynman's paper, I'm not aware of any extensive use of these ladder operators in any other model of quantum computation.

Is there any such model of quantum computation that focuses on ladder operators in lieu of unitary or hermitian operators? Did Feynman's "calculus" in his 1985 paper ever gain traction anywhere?

This is also partly motivated because I've read that Sophus Lie sort of envisioned infinitesimal generators of what we now call a Lie algebra as actual elements of the Lie group; it took others like Killing and Cartan to revise and formalize this intuition, and to put the Lie group - Lie algebra correspondence on more solid footing in the sense that the Lie algebra "linearizes" or is "tangent to" the Lie group.

I think perhaps there may be a similar analogy between the works of Feynman and, for example, Lloyd and Kitaev and Aharanov and others who came after, in the sense that these ladder operators somehow "linearize" or are "tangent to" the Hamiltonian, which enables Hamiltonian simulation. I'm trying to understand more of Feynman's intuition regarding these operators, but I might be making this analogy in hindsight.


1 Answer 1


Looking over Feynman's paper, the ladder operators he uses are spin ladder operators, what we would in modern language write as $\sigma_+$ and $\sigma_-$. (This is in contrast to fermionic or bosonic annihilation or creation operators, I'm assuming you are not asking about those.) Note that the operator that you've written down for the CCNOT, in this language, is a unitary matrix.

I guess what you are asking about is whether these operators, as used in a Lie algebraic sense, are used in quantum computing. In some sense they are always used because the $\sigma_\pm$ operators and $\sigma_+ \sigma_-$ are a basis for 2x2 matrices. So when we use $X$, $Y$, and $Z$ Pauli operators we could always express them in this basis.

But maybe you are after whether the Lie algebraic point of view is used. One place this occurs is in decoherence-free (noiseless) subspaces and subsystems. A good reference is this review article. In particular, looking at the Lie algebra of the system part of a system-bath interaction can give you information about subspaces that are protected from decoherence. For example if your bath couples to all your qubits in exactly the same way, you end up seeing terms like $\sum_i (\sigma_+)_i$ and $\sum_i (\sigma_-)_i$ which are best analyzed by looking at the Lie algebra.

By the way, one of the interesting parts about Feynman's paper is that he constructs a "clock state", this was later used by Kitaev and even later by others to show a lot of interested results in quantum complexity theory and adiabatic quantum computing.


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