56 PROC. OF THE 11th PYTHON IN SCIENCE CONF. (SCIPY 2012) QuTiP: A framework for the dynamics of open quantum systems using SciPy and Cython ‡ ‡ Robert J. Johansson ∗, Paul D. Nation QuTiP Organization 6/14/12 3:35 PM F Abstract—We present QuTiP (http://www.qutip.org), an object-oriented, open- three_level_basis three_level_ops source framework for solving the dynamics of open quantum systems. Written wigner in Python, and using a combination of Cython, NumPy, SciPy, and matplotlib, qfunc tensor liouvillian steadystate QuTiP provides an environment for computational quantum mechanics that about spost spre Bloch brmesolve bloch_redfield_tensor is both easy and efficient to use. Arbitrary quantum systems, including time- thermal_dmsteady clebsch qutrit_basis three_level_atom correlation spectrum_ss dependent systems, may be built up from operators and states defined by a projection demos ket2dm superoperator concurrence quantum object class, and then passed on to a choice of unitary or dissipative fock_dm entropy_conditional coherent_dm wigner entropy_linear tensor entropy_mutual evolution solvers. Here we give an overview of the basic structure for the fock entropy_vn coherent concurrence steady about Bloch bloch-redfield framework, and the techniques used in its implementation. We also present basis clebsch eseries correlation sphereplot demos a few selected examples from current research on quantum mechanics that states essolve entropy ode2es sp_eigs illustrate the strengths of the framework, as well as the types of calculations sphereplot eseries expect simdiag essolve that can be performed. This framework is particularly well suited to the fields of sparse file_data_read expect file_data_store rhs_generate simdiag rhs_generate fileio qload quantum optics, superconducting circuit devices, nanomechanics, and trapped qsave rand_unitary rand floquet ions, while also being ideal as an educational tool. rand_ket fmmesolve gates rand_herm qstate QuTiP cnot rand_dm fredkin graph Qobj phasegate Index Terms—quantum mechanics, master equation, monte-carlo qstate snot istests swap shape toffoli Qobj mcsolve propagatorparfor mesolve metrics hinton ptrace orbital Odedata dims Odeoptions dag isbra ischeck Introduction isequal isherm operators isket isoper issuper One of the main goals of contemporary physics is to control the propagatorparfor mcsolve orbital mesolve dynamics of individual quantum systems. Starting with trapped- propagator_steadystate odesolve fidelity squeez tracedist sigmaz Odedata Odeoptions sigmay create jz ion experiments in the 1970s [Hor97], the ability to manipulate sigmax sigmap qeye num sigmam jplus jmat qutrit_ops destroy single realizations, as opposed to ensembles, of quantum systems displace allows for fundamental tests of quantum mechanics [Har06] and quantum computation [Lad10]. Traditionally, the realm of quan- Fig. 1: The organization of user available functions in the QuTiP tum mechanics has been confined to systems with atomic and framework. The inner circle represents submodules, while the outer molecular characteristic length scales with system properties fixed circle gives the public functions and classes contained in each by nature. However, during the last two decades, advances in submodule. experimental and manufacturing techniques have opened up the possibility of producing man-made micro and nanometer scale devices with controllable parameters that are governed by the bath, where the complexity of the environmental dynamics renders laws of quantum mechanics. These engineered systems can now the combined evolution of system plus reservoir analytically be realized with a wide range of different technologies such intractable and must therefore be simulated numerically. With a as quantum optics [Obr09], superconducting circuits [You11], quantum computer out of reach for the foreseeable future, these semiconducting quantum dots [Han08], nanomechanical devices simulations must be performed using classical computing tech- [Oco10], and ion traps [Bla12], and have received considerable niques, where the exponentially increasing dimensionality of the experimental and theoretical attention. underlying Hilbert space severely limits the size of system that can With the increasing interest in engineered quantum devices, be efficiently simulated [Fey82]. Fortunately, recent experimental the demand for efficient numerical simulations of openfile:///Volumes/dml.riken.jp/public_html/qutip_org.html quantum advances have lead to quantum systems fabricated from a small Page 1 of 1 dynamics has never been greater. By definition, an open quantum number of oscillator and spin components, containing only a few system is coupled to an environment, also called a reservoir or quanta, that lend themselves to simulation in a truncated state space. * Corresponding author: [email protected] ‡ Advanced Science Institute, RIKEN, Wako-shi, 351-0198 Japan Here we introduce QuTiP [Joh12], a framework designed for simulating open quantum systems using SciPy and Cython. Al- Copyright © 2012 Robert J. Johansson et al. This is an open-access article though other software solutions exist [Sch97], [Vuk07], [Tan99], distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, QuTiP goes beyond these earlier tools by providing a completely provided the original author and source are credited. open-source solution with an easy to read syntax and extended QUTIP: A FRAMEWORK FOR THE DYNAMICS OF OPEN QUANTUM SYSTEMS USING SCIPY AND CYTHON 57 functionality, such as built-in multiprocessing. Our objective with numerical simulations of quantum systems on classical computers QuTiP is to provide a thoroughly tested and well documented is therefore an important subject. generic framework that can be used for a diverse set of quantum Although the state of an ideal quantum systems is completely mechanical problems, that encourages openness, verifiability, and defined by the wavefunction, or the corresponding state vector, reproducibility of published results in the computational quantum for realistic systems we also need to describe situations where the mechanics community. true quantum state of a system is not fully known. In such cases, the state is represented as a statistical mixture of state vectors yn , that can conveniently be expressed as a state (density) matrix Numerical quantum mechanics j i r = ∑ pn yn yn , where pn is the classical probability that the n j ih j In quantum mechanics, the state of a system is represented by system is in the state yn . The need for density matrices, instead the wavefunction Y, a probability amplitude that describes, for of wavefunctions, arisesj ini particular when modeling open quan- example, the position and momentum of a particle. The wavefunc- tum system, where the system’s interaction with its surrounding tion is in general a function of space and time, and its evolution is included. In contrast to the Schrödinger equation for closed is ideally governed by the Schrödinger equation, i¶t Y = Hˆ Y, − quantum systems, the equation of motion for open systems is not where Hˆ is the Hamiltonian that describes the energies of the unique, and there exists a large number of different equations possible states of the system (total energy function). In general, of motion (e.g., Master equations) that are suitable for different the Schrödinger equation is a linear partial differential equation. situations and conditions. In QuTiP, we have implemented many For computational purposes, however, it is useful to expand the of the most common equations of motion for open quantum wavefunction, Hamiltonian, and thus the equation of motion, in systems, and provide a framework that can be extended easily terms of basis functions that span the state space (Hilbert space), when necessary. and thereby obtain a matrix and vector representation of the system. Such a representation is not always feasible, but for many physically relevant systems it can be an effective approach when The QuTiP framework used together with a suitable truncation of the basis states that As a complete framework for computational quantum mechanics, often are infinite. In particular, systems that lend themselves to QuTiP facilitates automated matrix representations of states and this approach includes resonator modes and systems that are well operators (i.e. to construct Hamiltonians), state evolution for characterized by a few quantum states (e.g., the two quantum closed and open quantum systems, and a large library of common levels of an electron spin). These components also represent the utility functions and operations. For example, some of the core fundamental building blocks of engineered quantum devices. functions that QuTiP provides are: tensor for constructing In the matrix representation the Schrödinger equation can be composite states and operators from its fundamental components, written as ptrace expect d for decomposing states into their components, i y = H(t) y ; (1) for calculating expectation values of measurement outcomes for − dt j i j i an operator and a given state, an extensive collection of functions where y is a state vector and H is the Hamiltonian matrix. for generating frequently used states and operators, as well as j i Note that the introduction of complex values in (1) is a fun- additional functions for entanglement measures, entropy measures, damental property of evolution in quantum mechanics. In this correlations and much more. A visual