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One scaling framework for three Lindblad limits

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This note shows how three microscopic derivations of a Markovian Lindblad (GKSL) generator are really just variations on the same basic framework, differing in the relative strengths of the system, bath, and interaction Hamiltonians:

  • the weak-coupling/Davies limit (Born–Markov plus a secular average),
  • the singular-coupling/white-noise limit (fast bath with appropriately rescaled coupling), and
  • the fast-system limit (“super-secular”).

The first two are familiar. The third is partially degenerate, and so less often discussed, but appears, e.g., in the classic decoherence literature on chiral molecules. The three variations are often presented in ways that make them look completely distinct, which obscures their close relationship.

The scaling ansatz and the Markov limit

Fix a system–bath decomposition and consider a one-parameter family of Hamiltonians

(1)   \begin{align*} H^{(\lambda)}=\lambda^{-a}H_S+\lambda^{-c}V+\lambda^{-b}H_B, \qquad \lambda\to 0, \end{align*}

where H_S acts on the system Hilbert space, H_B on the bath Hilbert space, and V couples them. Think of a,b,c \in \mathbb{R} as encoding how quickly the system rotates (\lambda^{-a}), how quickly the bath decorrelates (\lambda^{-b}), and how strongly the coupling acts (\lambda^{-c}), all while keeping the physical time variable t fixed.

We assume the usual minimal structure used in essentially all microscopic derivations of the Lindblad equation:

  • Factorized initial state: \rho(0)=\rho_S(0)\otimes \eta_B.
  • Stationary bath state: [\eta_B,H_B]=0.
  • Centered coupling: \operatorname{Tr}_B(\eta_B V)=0, i.e., no first-order drift.

A convenient form is

(2)   \begin{align*} V=\sum_\alpha A_\alpha\otimes B_\alpha, \qquad \operatorname{Tr}_B(\eta_B B_\alpha)=0. \end{align*}

The question is: for which (a,b,c) can the reduced dynamics

    \[\rho_S^{(\lambda)}(t)=\operatorname{Tr}_B\!\Big(e^{-iH^{(\lambda)}t}\big(\rho_S(0)\otimes\eta_B\big)e^{iH^{(\lambda)}t}\Big)\]

have a nontrivial, finite \lambda\to 0 limit that is a time-homogeneous quantum dynamical semigroup, i.e., a Lindblad equation?

Why the scaling constraint 2c = b is essentially forced

Go to the interaction picture with respect to the free Hamiltonian

    \[H_0^{(\lambda)}=\lambda^{-a}H_S+\lambda^{-b}H_B.\]

Then the interaction-picture coupling is

    \[V_I^{(\lambda)}(t)=\lambda^{-c}\sum_\alpha A_\alpha^{(\lambda)}(t)\otimes B_\alpha^{(\lambda)}(t),\]

with

    \[A_\alpha^{(\lambda)}(t)=e^{i\lambda^{-a}H_S t}A_\alpha e^{-i\lambda^{-a}H_S t},\qquad B_\alpha^{(\lambda)}(t)=e^{i\lambda^{-b}H_B t}B_\alpha e^{-i\lambda^{-b}H_B t}.\]

Bath correlations in the stationary state \eta_B are

    \[C_{\alpha\beta}(t)=\operatorname{Tr}_B\!\big(\eta_B B_\alpha(t)B_\beta(0)\big),\qquad B_\alpha(t)=e^{iH_B t}B_\alpha e^{-iH_B t}.\]

Because B_\alpha^{(\lambda)}(t)=B_\alpha(t/\lambda^b), one has

(3)   \begin{align*} \operatorname{Tr}_B\!\big(\eta_B B_\alpha^{(\lambda)}(t)B_\beta^{(\lambda)}(s)\big)=C_{\alpha\beta}\!\left(\frac{t-s}{\lambda^b}\right). \end{align*}

Now recall what every Born–Markov style derivation is doing at second order: the reduced generator is built from a memory integral of the form

(4)   \begin{align*} \frac{d}{dt}\rho_S^{(\lambda)}(t)\;\approx\;-\int_0^\infty ds\;\operatorname{Tr}_B\!\Big([V_I^{(\lambda)}(t),[V_I^{(\lambda)}(t-s),\rho_S^{(\lambda)}(t)\otimes \eta_B]]\Big), \end{align*}

where “\approx” means

  1. truncate at second order in the interaction,
  2. replace \rho_S^{(\lambda)}(t-s) by \rho_S^{(\lambda)}(t) inside the integral, and
  3. extend the upper limit to \infty using the decay of bath correlations.
… [continue reading]
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LLM-optimized research articles

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Summary: Assume there are several years when AI assistants (“LLMs”) have super-human ability to understand and apply known ideas, but sub-human ability to generate genuinely new ones. In this window, research communication should shift substantially. Rather than writing papers optimized for human readers, we should expect researchers will explain ideas conversationally to an LLM, which produces a written artifact, which audience members then consume through their own LLM. This combines the adaptability of one-on-one communication with the scalability of text. The written artifact probably remains human-readable—for training stability, peer review, and keeping humans in the loop—but its conventions would drift toward an LLM-optimized style: high information density, negligible review of existing knowledge, greater length variability, non-linear organization, exhaustive referencing, and explicit uncertainty markers.

Premise

A cartoon history of communication might use audience size as a key organizing metric:

  1. One-to-one (e.g., conversation): The oldest format. Even today, this is generally the most efficient way to communicate an idea to a specific person because it is adaptable. The listener can ask the speaker to speed up or slow down, or to explain points they don’t understand. But it doesn’t scale; the process must be repeated for each new listener.
  2. One-to-few (e.g., lectures, speeches): These formats scale better, but as the audience grows, individual listeners can ask fewer clarifying questions, and speakers can less easily tailor the message to what the audience already knows. Personal tutors are substantially more effective than classroom teachers.
  3. One-to-many (e.g., written documents, recorded video): These formats scale nearly perfectly because copying information is ~free, but efficiency falls substantially: there is no interactive feedback. (Readers can contact the author, but this reduces to one-on-one communication and doesn’t scale.)
… [continue reading]
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Hydrodynamical variables associated with approximately conserved quantities

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Wavefunction branches are probably approximate eigenstates of hydrodynamic variables (and maybe more generally of slow local variables). Hydro variables are local averages of approximately locally conserved quantities, not restricted to fluids. They are most useful when the remaining degrees of freedom are all able to locally thermalize quickly conditional on the hydro variables, so that hydro variables follow closed equations of motion (with noise, generally described by a Lindbladian) that are derived by treating the remaining degrees of freedom as a heat bath.

Although the most famous hydro variables corresponds to exactly conservedI will say “conserved” rather than “locally conserved” for brevity, but local conservation — that they obey a continuity equation — is crucial for ensuring their slow behavior relative to the microscopic other degrees of freedom.a   quantities like energy and momentum, there are many systems where approximately locally conserved quantities are the natural way to describe macroscopic behavior. This post just collects a few examples:

  • Center-of-mass position for rigid object: I think there may be a few ways to characterize this, but for a system composed of atoms bound into rigid objects, the hydro variables are the densities of momentum (which is exactly conserved) and particle number (which is approximately conserved, and which maps to the location and orientation of objects), along with additional constraints imposed by the atomic bonds in the object. Particle number and the atomic bonds are only approximately conserved because bonds can break, chemical reactions can take place, nuclei can decay, etc.
  • Chemical species concentration during slow reactions: Consider a reactive flow, i.e., a fluid composed of molecules that are chemically reacting. If all reaction timescales for a certain chemical species are much longer than the timescale of the microscopic dynamics, then the concentration of that species will be approximately conserved.
… [continue reading]
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Checking our forecast for quantum computing in 2024

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Scott Aaronson has a nice post discussing recent experimental progress in quantum computing from Google, Microsoft, and Quantinuum. In particular, he says

Let me end by sticking my neck out. If hardware progress continues at the rate we’ve seen for the past year or two, then I find it hard to understand why we won’t have useful fault-tolerant QCs within the next decade. (And now to retreat my neck a bit: the “if” clause in that sentence is important and non-removable!)

If you’ll permit me to indulge, I’d like to compare the reported progress to the predictions from the 2020 forecasting paper by Jaime Sevilla and me.

I understand the Google results to be that they have ~10-3 physical two-qubit error rates and they they would have ~10-6 logical error rates if they had ~1500 physical qubits, but they actually have 101 in the present device.

For forecasting purposes, Jaime and I defined a figure of merit, the “generalized logical qubit” (GLQ) count. See Eq. (1) and Fig. 1. It essentially equals the surface-code logical qubit count in the limit of low errors and many qubits (assuming we demand 10-18 logical error rate), but is smoothly defined with fractional values when error rates are larger and/or physical qubit counts are smaller. Our naive (log linear regression) forecast from data at the time was that 1 GLQ would be achieved 2026-2033 and 4,100 GLQs (the rough number necessary to be cryptographically dangerous) would be achieved in 2029-2060 (both 90% confidence intervals — not Bayesian credences!). Our forecasts for 2024 were 0.02-0.3 GLQs.

I’m pleased to see that the above Google numbers correspond to a GLQ count of about 0.03, which is consistent with our model (though this is perhaps not impressive since it’s only been 4 years and our confidence intervals were wide).… [continue reading]

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Generalizing wavefunction branches to indistinguishable subspaces

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[This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.]

An original dream of defining branches based on redundant records (aka redundant classical information, aka GHZ-like correlations) was that it would be possible to decompose the wavefunction of an evolving non-integrable quantum system at each point in time into macroscopically distinguishable branches that individually had bounded amounts of long-range entanglement (i.e., could be efficiently expressed as a matrix product state) even though the amount of long-range entanglement for the overall state diverges in time. If one could numerically perform such a decomposition, and if the branches only “fine-grain in time”, then one could classically sample from the branches to accurately estimate local observables even if the number of branches increases exponentially in time (which we expect them to do).

However, we now think that only a fairly small fraction of all long range entanglement can be attributed to redundantly recorded branches. Thus, even if we found and efficiently handled all such classical information using a decomposition into a number of branches that was increasing exponentially in time (polynomial branch entropy), most branches would nevertheless still have an entanglement entropy across any spatial partition that grew ~linearly in time (i.e., exponentially increasing bond dimension in the MPS representation) until saturating.

In this post I’ll first write down a simple model that suggests the need to generalize the idea of branches in order to account for most long-range entanglement. Then I will give some related reasons to think that this generalized structure will take the form not of a preferred basis, but rather preferred subspaces and subsystems, and together these will combine into a preferred “branch algebra”.… [continue reading]

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What logical structure for branches?

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[This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.]

Taylor & McCulloch have a tantalizing paper about which I’ll have much to say in the future. However, for now I want to discuss the idea of the “compatibility” of branch decompositions, which is raised in their appendix. In particular, the differences between their approach and mine prompted me to think more about how we could narrow down on what sorts of logicalThis is “logic” in the same sense of identifying sets of propositions in consistent histories that comport with the axioms of a classical probability space, before discussing any questions of physics.a   axioms for branches could be identified even before we pin down a physical definition. Indeed, as I will discuss below, the desire for compatibility raises the hope that some natural axioms for branches might enable the construction of a preferred decomposition of the Hilbert space into branching subspaces, and that this might be done independently of the particular overall wavefunction. However, the axioms that I write down prove to be insufficient for this task.

Logical branch axioms

Suppose we have a binary relation “\perp\hspace{-1.1 em}\triangle” on the vectors in a (finite-dimensional) Hilbert space that indicates that two vectors (states), when superposed, should be considered to live on distinct branches. I will adopt the convention that “z = v\perp\hspace{-1.1 em}\triangle w” is interpreted to assert that z=v+w and that the branch relation v\perp\hspace{-1.1 em}\triangle w holds.This doesn’t constrain us because if we just want to assert the binary relation without asserting equality of the sum to a third vector, we write v\perp\hspace{-1.1 em}\triangle w without setting it equal to anything, and if we just want addition without asserting the relation, we write v+w=z.… [continue reading]

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Branching theories vs. collapse theories

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This post explains the relationship between (objective) collapse theories and theories of wavefunction branching. The formalizations are mathematically very simple, but it’s surprisingly easily to get confused about observational consequences unless laid out explicitly.

(Below I work in the non-relativistic case.)

Branching: An augmentation

In its most general form, a branching theory is some time-dependent orthogonalSome people have discussed non-orthogonal branches, but this breaks the straightforward probabilistic interpretation of the branches using the Born rule. This can be repaired, but generally only by introducing additional structure or principles that, in my experience, usually turns the theory into something more like a collapse theory, which is what I’m trying to constrast with here.a   decomposition of the wavefunction: \psi= \sum_{\phi\in B(t)} \phi where B(t) is some time-dependent set of orthogonal vectors. I’ve expressed this in the Heisenberg picture, but the Schrödinger picture wavefunction and branches are obtained in the usual (non-branch-dependent) way by evolution with the overall unitary: \psi(t)=U_t \psi and \phi(t)=U_t \phi.

We generally expect the branches to fine-grain in time. That is, for any two times t and t'>t, it must be possible to partition the branches B(t') at the later time into subsets B(t',\phi) of child branches, each labeled by a parent branch \phi at the earlier time, so that each subset of children sums up to its corresponding earlier-time parent: \phi = \sum_{\phi' \in B(t',\phi)} \phi' for all \phi\in B(t) where B(t') = \bigcup_{\phi\in B(t)} B(t',\phi) and B(t',\phi)\cap B(t',\tilde\phi) for \phi\neq\tilde\phi. By the orthogonality, a child \phi' will be a member of the subset B(t',\phi) corresponding to a parent \phi if and only if the overlap of the child and the parent is non-zero. In other words, a branching theory fine-grains in time if the elements of B(t) and B(t') are formed by taking partitions P(t) and P(t') of the same set of orthogonal vectors, where P(t') is a refinement of P(t), and vector-summing each subset of the respective partition.… [continue reading]