  Superconducting quantum circuits is one of the leading candidates for a
universal quantum computer. Designing novel qubit and multiqubit
superconducting circuits requires the ability to simulate and analyze the
properties of a general circuit. In particular, going outside the transmon
approach, we cannot make assumptions on anharmonicity, thus precluding blackbox
quantization approaches and necessitating the formal circuit quantization
approach. We consider and solve two issues involved in simulating general
superconducting circuits. One of the issues is the handling of free modes in
the circuit, that is, circuit modes with no potential term in the Hamiltonian.
Another issue is circuit size, namely the challenge of simulating strongly
coupled multimode circuits. The main mathematical tool we use to address these
issues is the linear canonical transformation in the setting of quantum
mechanics. We address the first issue by giving a provably correct algorithm
for removing free modes by performing a linear canonical transformation to
completely decouple the free modes from other circuit modes. We address the
second by giving a series of different linear canonical transformations to
reduce intermode couplings, thereby reducing the problem to the weakly coupled
case and greatly mitigating the overhead for classical simulation. We benchmark
our decoupling methods by applying them to the circuit of two inductively
coupled fluxonium qubits, obtaining several orders of magnitude reduction in
the size of the Hilbert space that needs to be simulated.
