-0.4 -0.2 0 0.2 0.4 0.6 t γ Amplitude H0(t) f(t) g(t)
Figure III.3 |Plot of the dilation for a qubit subjected to an amplitude damping Lindbla- dian and a resonant sinusoidal driving field. The real part of the different components of the Hamiltonian detailed in Eq.(III.46) are shown here with ω0/γ = 2. Only one of the terms diverges at t = 0, while all the terms
decay to 0 quickly at large times.
III.E
Conclusion
A core method for constructing the dilation of a family of CPT maps to a time-dependent Hamiltonian acting on the system and a finite dimensional ancilla was presented. This was demonstrated on two typical examples of qubit noise, dephasing and amplitude damping, and gave simple Hamiltonians on two qubits in both the Markovian and non-Markovian case. The case of extra rotations on the system was also investigated perturbatively. These all showed that the dilation Hamiltonian was well behaved everywhere but at t = 0, a consequence of needing to circumvent the quantum Zeno effect. Truncating the divergent Hamiltonian to a fixed maximum value however had negligible impact on the total dynamics.
By rescaling time in a nonlinear way, some of these channels can even be dilated to constant Hamiltonians. In general, this is true whenever the dilation Hamiltonian is of the form H(t) = h(t)X such that it has a single time-dependent factor. This allows a constant Hamiltonian H = h0X to be applied for time τ to simulate the real dynamics evolving for
a time t where τ = h1
0
Rt
0h(t0)dt0. As h(t) is continuous and bounded, this is always well
defined. In cases where there are several different time dependencies, this method can be used to remove one of them. This is particularly useful in eliminating divergences, which would otherwise be problematic to implement experimentally. It also has the advantage that, in many cases, the evolution for an infinite amount of time t can be simulated with a finite τ .
As well as being useful in the single qubit case, these results can also be directly used in the case of a system of many qubits all subjected to independent channels, leading to a
CHAPTER III. MINIMAL DILATIONS
super-polynomial speed up from what could be achieved classically. Prominent questions which could be investigated include the decay of multipartite entanglement of an initially highly entangled state [LM13], or the performance of one way computation [RB01] with a cluster state affected by local noise. For such systems of N qubits the dilation can be calculated once classically and the dynamics of the entire system then simulated on a quantum computer using a maximum of 2N ancilla qubits and Hamiltonians that affect, at most, three qubits. Simulating this classically would require applying the quantum channel up to 2N times for an initial state which is highly entangled or, equivalently, solving the dynamics of the complete channel which would have 2N Kraus operators.
This method also provides the ability to do something which a normal Stinespring dilation cannot do at all. By simulating the dynamics continuously in time, the evolution of the state follows the ‘true’ path that is being simulated in the Hilbert space, rather than just reaching the required goal for a single instance in time. This means that the full information about the behaviour of the system over an extended interval of time is accessible, allowing simulations where the time at which measurement takes place is not known a priori. In our scheme, such a scenario creates no additional difficulty, as the sys- tem follows the correct continuous dynamics. In a standard Stinespring dilation approach, however, this is either completely impossible or introduces substantial errors. These can be quantified by considering the snapshot Stinespring dilation as being a constant Hamil- tonian (the logarithm of the unitary dilation) applied for different durations, leading to an error which grows in time.
In our approach, the complexity of open system dynamics is condensed in the time- dependent system-environment interaction, allowing a simulation to be implemented us- ing state of the art methods; whereas in natural systems, and in previous approaches to simulating the dynamics of open systems, the complexity resides in the dynamics of the infinitely large environment [Gre01, ZˇSB05, KBA05]. These two perspectives can be understood as the two ends of the spectrum of quantum simulations. Since any time- dependence can be understood as originating from the dynamics of an additional system (fundamentally all interactions are time-independent), the approached described in this chapter suggests very clearly how to access the entire spectrum: expand the ancilla sys- tem while gradually reducing the time-dependence of the interaction. A general proce- dure for doing this is the clock-construction [MSK15]. Such a continuous variation will give valuable insight, for example in the controllability of open quantum dynamics, as seen in Eqs.(III.45-III.46). There the back-action of an external control caused by the environment interaction makes itself transparent, whereas such effects are extremely hard to unravel in a model based on an infinitely large environment, as can be seen in §IV. The suggested transition would allow the study of this back-action in its entire range of manifestations, opening up a completely new angle on the investigation of control on open quantum systems.
CHAPTER IV
LINDBLADIAN AND CONTROL
COMMUTATIVITY CONJECTURE
IV.A
Introduction
This idea for this chapter came up during the Quantum Information Processing Confer- ence (QIPC Leeds, 2015) where we were discussing with Rafal Demkowicz-Dobrzanski how, in control theory, Hamiltonians are often naively added on top of a Lindbladian without considering how it might modify the dissipative dynamics, as was done in §II. He suggested that it ought to be valid to take this approach if the control Hamiltonian and the Lindbladian commute, which we call the Lindbladian and Control Commutativity (LCC) conjecture. This chapter details our efforts in proving whether this is true or not.
IV.A.1 Motivation
In §II, no-go theorems for which CPT maps could be reached by an open system using coherent controls were derived, under the assumption that the addition of controls did not modify the dissipative part of the dynamics. However, in §III, it was clear that adding controls on top of the noise leads to a different dilation. This is a particular example of something which is well known (if commonly ignored): adding an extra Hamiltonian to the Schr¨odinger equation describing the unitary evolution of the system and the bath, and then reducing this to a Lindblad type equation of motion is not equivalent to reducing the Schr¨odinger equation and then adding the same Hamiltonian to the system [DJR14]. The two different ways of getting to the reduced dynamics in the presence of controls are shown diagrammatically in Fig.IV.1.
Naively adding the control Hamiltonian on top of the Lindbladian (the clockwise path in Fig.IV.1) is often the method taken, as in §II, although the opposite order would be more rigorous. The principle reason for doing this is that it is simpler both mathematically and physically. Tracing out the bath in order to obtain a Lindblad type master equation,
CHAPTER IV. LINDBLADIAN AND CONTROL COMMUTATIVITY CONJECTURE
as was done in §I.B.4, is hard and may not be tractable if complex dynamics happen to the system [DS78, DJR14] because this may cause Markovianity to be lost leading to the reduced system not obeying a simple Master equation. Having to redo this calculation for every potential control of interest would rarely be feasible; the very reason for deriving a master equation is to be able to abstract away the bath as much as possible. Furthermore, the noise operator in the absence of control is often known (approximately) experimentally by observing how the system decays. It is harder, and not always possible, to calculate from this the exact interaction Hamiltonian between the system and the bath [Cyw14, Z ´AK16]. For these reasons, it is desirable to know when the two paths in Fig.IV.1 are (approximately) equivalent. It is already known that in the singular coupling limit this is a valid approximation [DJR14], but in many cases it is not.
H
SBL + H
C?
L
H
SB+ H
CFigure IV.1 |Flowchart of the two different ways of adding controls to a Lindblad master equation. The starting point is always the exact Hamiltonian describing the system and bath. In the anticlockwise path the control Hamiltonian is added to it, and the bath is then traced out to give a reduced equation of motion on the system that incorporates the controls. It is however often easier and desirable to take the clockwise the path, where the bath is first traced out, giving a fixed Lindbladian for the system, and the control Hamiltonian then added. In general these two approaches give different answers: the order of the operations do not commute.
An interesting case when the two paths of Fig.IV.1 are not equivalent, and even max- imally different, is in the Shallow-Pocket model [AHFB15, ABFH17]. In this case the use of dynamical decoupling type of controls [VKL99, LB13] in the clockwise path leads to periodic rotations on top of decoherence, while in the anti-clockwise path the controls undo the effect of the noise. The reason for this can be seen by solving the model. It consists of a qubit interacting with a bath made up of a single free particle in 1-D via the Hamiltonian Hs-p = 12gσz⊗ x. If the free particle is originally in the state |ψi where
hx|ψi =r γ π
1
IV.A. INTRODUCTION
the reduced dynamics on the qubit are L(ρ) =−γ
4g[σz, [σz, ρ]], (IV.2) which corresponds exactly to the dephasing Lindbladian of§I.B.6 where the coherence of the qubit decays exponentially in time. The controls used are an instantaneous σxrotation
on the qubit, exp(−iπ
2σx) = σx, applied at fixed intervals of τ . If such controls are applied
in the reduced picture the final state is
ρ(2τ ) = σx eLτσxeLτ(ρ(0))σx σx (IV.3)
= eLτeLτ(ρ(0)) , (IV.4)
because
eLτ (σx· σx) = σxeLτ(·) σx, (IV.5)
therefore the evolution of the qubit after time 2τ is the same as not applying any controls. Doing this in the microscopic picture, however, completely removes the decoherence as the same control sequence acts as [AHFB15]
(σx⊗ IB) e−iHs-pτ (σx⊗ IB) e−iHs-pτ = eiHs-pτ e−iHs-pτ (IV.6)
= I. (IV.7)
This shows how drastic the difference between the two different ways of adding controls can be.
This sort of calculation, however, relies on knowing the precise interaction between the system and the bath. The aim of this chapter is to see what can be said when this is not known, but extra constraints on the allowed controls are imposed. The Lindbladian and Control Commutativity (LCC) conjecture is that if the control Hamiltonian commutes with the Lindbladian on the system at all times, then it is exactly valid to add the control on top of the noise no matter what the microscopic interaction between the system and the bath is. This conjecture was suggeted by Rafal Demkowicz-Dobrzanski, who has used it in some of his work [SKHDD16]. The intuition for why this should be true is because if the controls and noise commutes then it is possible to consider the controls acting instantaneously after the evolution under the Lindbladian. In this case, the noise is ‘unaware’ that there will be controls and so should remain unchanged. A rigorous proof of this however has alluded us and we have been unable to prove or disprove this conjecture either way. The heart of the problem is that, although the controls commute with the Lindbladian, they may not commute with the microscopic system-bath interaction, so it is not necessarily valid to consider the two acting separately.
CHAPTER IV. LINDBLADIAN AND CONTROL COMMUTATIVITY CONJECTURE
where several of our attempts to prove it result in the partial results given. In §IV.C a particularly interesting model is investigated which is numerically shown to violate the conjecture for some choice of parameters, yet is analytically proved to obey it in the Markovian limit. Finally, this chapter is concluded in §IV.D with a summary of our results, along with their implications and limitations.
IV.A.2 Problem Definition
The conjecture involves a finite dimensional system ρS whose evolution in the absence of
controls is effectively described by a Lindbladian L. This arises from a time-independent interaction with a bath
ρS(t) = eLt(ρS) (IV.8)
= TrBe−iHSBt(ρS⊗ ρB)eiHSBt
(IV.9) = TrBeHSBt(ρS⊗ ρB) , (IV.10)
where HSB is the complete system-bath Hamiltonian, HSB is its adjoint action (H(·) =
−i[H, · ]), and ρB is the initial state of the bath. Although L is known, HSB and ρB are
not, and there are generally a great many different dilations that give the same reduced dynamics on ρS. The conjecture states that given an additional Hamiltonian HC on the
system alone, such that [HC, L] = 0, then, for all {HSB, ρB} that satisfy Eq.(IV.10):
e(L+HC)t(ρ S) = TrB h e(HSB+HC⊗I)t(ρ S⊗ ρB) i , (IV.11)
for all t and all initial ρS. In the case that the controls are time-dependent, the equivalent
statement is T eR0tL+HC(τ )dτ(ρ S) = TrB h T eR0tHSB+HC(τ )⊗Idτ(ρ S⊗ ρB) i . (IV.12) For convenience, the shorthandHC ⊗ I = H0C is used below.
There are two observations worth making at this point. Firstly, the conjecture is true if [HC0 , HSB] = 0. This makes intuitive physical sense, if the controls commute with the
microscopic interaction between the system and the bath, then the two ignore each other. Therefore any attempts to disprove the conjecture, or find a counter example, must rely on the control not commuting with the system-bath Hamiltonian even though it commutes with the Lindbladian. However, the result can hold even in such a case; a toy model is detailed in§IV.C where the controls do not commute with the full Hamiltonian, but the LCC conjecture holds.
Secondly, for L to be a true Lindbladian it is necessary for HSB to be unbounded,