# The quantum ENIAC and Schrodinger’s glass

Language is wonderfully subtle. Two sentences can mean the same thing but leave the reader with completely different impressions. Everyone can appreciate that “The glass is half empty” and “The glass is half full” actually mean the same thing. Yet, they leave different impressions on the reader.

Sabine Hossenfelder’s recent Guardian column on quantum computation is titled

Quantum supremacy is coming. It won’t change the world

and it muses on the impending announcement of quantum supremacy. Soon, someone (possibly Google) is going to announce that they have used a quantum computer to perform a particular calculation faster than our fastest conventional supercomputers! Sabine’s spin is that Schrodinger’s glass is very much, half-empty. Even worse it is half empty with gone off milk.

Should you believe her?

We are in the midst of a surge of optimism, investment and frequent hype around quantum computation. Sabine’s column is a refreshing change to the worst of the hyperbole currently circulating. It has a similar, acerbic tone that she also takes in her critique of particle physics. It is fun to read and I’ve enjoyed many of her blog articles on the disconnect between experiment and theory in the high energy physics community. Of course, a balanced perspective sits somewhere between the extremes of Sabine’s pessimism and those overpromising on quantum computation.

Quantum supremacy is coming. It won’t change the world tomorrow

then she is absolutely right. The day after you won’t notice any difference.

But she insinuates that quantum computers will never change the world, which is unwarranted and absurd given the parallels she draws with early conventional computers. She compares the coming generation of quantum computers with an early computer ENIAC that was publically announced in 1946. It did take about 35 years from ENIAC until people could have their own personal computer (PC) at home, and 60 years until most people started carrying smart phones in their pockets! This is arguably the most rapid and significant change in human history, so the comparison is a bit bizarre.

It is difficult to see why Sabine makes the comparison with ENAIC, maybe the point is that many people are promising useful quantum computers in less than 35 years. But long before you got your first PC or smartphone, computers were having a huge (though often invisible) effect on our lives. Sabine suggests that ENIAC itself had no impact itself. However, before the public announcement of ENAIC’s existence, it was used to target artillery and also to design the first atomic bomb. Around this time, simple computing devices also played a crucial role in the growing telephone network. If 2019 was the year of the quantum-ENAIC and history repeats itself, then this would be a monumental event. As many others have pointed on the comments section of her article, the analogy with ENAIC is both flawed and fails to support her main headline.

Let’s set aside historical analogies. What else does Sabine say:

Quantum supremacy will be a remarkable achievement for science. But it won’t change the world any time soon. The fact is that algorithms that can run on today’s quantum computers aren’t much use. One of the main algorithms, for example, makes the quantum computer churn out random numbers. That’s great for demonstrating quantum supremacy, but for anything else? Forget it.

Here Sabine is referring to a specific task that many companies are planning to use as their first test of how the quantum computer performs against a conventional computer. She makes a strong point here. The so-called cross-entropy benchmarking tests are just as dull and technical as the name sounds.

To me, this is a benchmarking test. When a quantum device passes the test, it tells you that technology has reached a new level of maturity. Sabine is also right that cross-entropy benchmarking does not have any practical applications and no-one would pay a billion pounds for a device that could only perform this task. However,…

What the cross-entropy benchmarking test really shows is that the technology is sufficiently mature that it is worth running other programmes on the hardware. Much more exciting is to use a quantum computer solve problems in chemistry and material science. Computing power is a definite bottle-neck point in the design of drugs and materials, so we’d like to try and solve these problems on a quantum computer. In the near future (soon after passing the cross-entropy benchmarking tests) we can start our first attempts using a heuristic algorithm called VQE (the variational quantum eigensolver). Possibly, the 50-100 qubit devices coming soon would provide commercially useful solutions in this sector.

Why am I hedging with words like “possibly” and what does “heuristic algorithm” mean. I want to give a balanced perspective, so let explain what I mean here. First, let’s check the Wikipedia page for heuristic

The objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. This solution may not be the best of all the solutions to this problem, or it may simply approximate the exact solution. But it is still valuable because finding it does not require a prohibitively long time.

This doesn’t sound too bad. Indeed, the vast majority of conventional computing software is built on a mountain of heuristics so they are pretty useful. However, often it is hard to predict the performance of a heuristic until you’ve run it on an actual device. However, we can design heuristics and test them for 20-30 qubit on our laptop using an emulation of a quantum computer. So for these small sizes, your laptop could predict exactly what the quantum computer would do when you run the heuristic. This gives you a decent idea what will happen when you run the heuristic on a 50 qubit quantum computer.

This is a design process happening now, for instance by the company RIVERLANE that I work for part-time. But there is some uncertainty in the extrapolation from 20-30 qubits to 50-100. This all tells us that there is a chance to solve some exciting chemistry problems in the near-term, but the technology might need to be more mature before we see any big payoffs.

Why use heuristics? In the near future, quantum computers won’t be perfect and as Sabine notes “The trouble is that quantum systems are exceptionally fragile”. The longer the computation, the more the noise accumulates. Consequently, we are currently limited to short, heuristic computations.

There are many quantum algorithms where we can give a mathematical proof of how the quantum computer will perform! This sounds better than using a heuristic. But these algorithms are much bigger computations than the heuristics that I mentioned above. In these larger-scale algorithms, the noise accumulation problem can be avoided by continually correcting errors. Though this quantum error correction procedure does use up some of our qubits, further adding to the required scale of the device.

Okay, so how many good quality qubits would I need so that I can say with 100% certainty that the quantum computer would change the world. Here Sabine says:

To compute something useful – say, the chemical properties of a novel substance – would take a few million qubits. But it’s not easy to scale up from 50 qubits to several million.

A few million qubit is indeed what is estimated in some of the recent literature (see for example). There are two issues with this quote. First, it ignores the heuristic methods I mentioned earlier. Second, it suggests that it would be impossible to solve this problem with fewer than a million qubits! The number of qubits needed will depend on the quality of the hardware and the design of the algorithms and error correction methods. Come up with better designs of hardware or software and the numbers go down and the technology progresses. Only a few years ago, to solve the same chemistry problem our best estimates would have been closer to a billion qubits!

In the one million qubit blueprint, many of the qubits are busy removing noise. This would allow the quantum computer to run long enough to compute reliably. Sabine does not seem aware of this point as she says that

Even when housed in far-flung shelters and accessed through the cloud, it is not clear that a million qubit machine would ever remain stable for long enough to function.

She is missing the point that a million qubits are budgeted for precisely to provide a guarantee that it will remain stable for long enough. On the other hand, for the smaller 50-100 qubit computer, the stability issue is more nuanced because everything is heuristic and without error correction.

The technological and algorithmic obstacles are tough and challenging. But as we work hard on these problems, we see two things happen: hardware developers release better devices with more qubits and less noise; theorists come up with better designs that need fewer qubits. This is happening now, we are not stuck on any insurmountable problems.

Towards the conclusion of Sabine’s column she ponders

Researchers need to move on. They must take a good, hard look at the quantum computers they are building and ask how will they make them useful.

Again, Sabine states something factually correct but in a glass-half-empty tone. However, I am happy to report that this is exactly what we are doing. I have just got home from QEC2019, a 1-week conference in London (more info here) where we focused on how to make the quantum error correction components more efficient. The conference had theorists reporting new techniques and ideas for how to perform error correction. It also had experimentalists describing the latest hardware progress. Most people at QEC are pretty realistic and can see that the glass definitely contains 50% water / 50% air and the tap is still running.

# Quantum dynamics goes Monte Carlo!

This post is a light summary of a project just published in Physical Review Letters. I am very pleased that this work got selected as an editor’s suggestion and that the legendary Dominic Berry was commissioned to write a Physics view-point. His viewpoint really is a superb summary and you should go read it now, then come back here!

on how best to simulate time evolution using a quantum computer, which can also be helpful with other tasks like finding the ground state energy of a Hamiltonian. The standard method is to Trotterize the dynamics, and in some situations this works quite well, but it struggles with Hamiltonians containing a large number of terms.

If a Hamiltonian has L terms then even the best known Trotter methods have a gate count scaling quadratically with L. This is especially problematic in molecular systems because an N electron/qubit system will have about N^4 terms in the Hamiltonians, so L^8 gates in the Trotterized approach. Since systems beyond the reach of classical computers have N>40, the numbers quickly become horrifying.

My new work proposes a protocol with no explicit L dependence in the gate count, so it performs much better for Hamiltonians with many, many terms. The key idea is to use randomness.

Before getting to the new ideas, let’s recap Trotterization using some simple pictures. In Trotterization, we divide the evolution into r Trotter steps. In the simplest version of Trotter, each step contains one gate for each term in the Hamiltonian as follows.

Each gate is a coloured block with the width indicating the duration of the pulse. The width of the blocks are proportional to the corresponding coefficients in the Hamiltonian. In the standard cost model, each of these blocks has unit cost, independent of the duration of the gate, so the above sequence has cost 5. Next, we repeat this r times to obtain a sequence illustrated by:

where here we have taken 10 Trotter steps. There are 5 gates per step and so 50 gates in the whole sequence. The black ticks are just to guide the eyes to the divisions between each step.

How to do better? The main idea is to use random compiling. Both me and Matt Hasting showed a couple of years ago that randomisation can be very useful in compiling quantum circuits. If your interested in this more general problem, I gave a talk at QIP at Delft. Childs, Ostrander, and Su have a paper where they suggest randomly permuting gates within each Trotter step, so for a particular sample you might get a sequence like

This does lead to improved performance but does not escape the L^2 dependence of the gate count, so still isn’t especially suited to Hamiltonians with very many terms.

In my paper, I proposed that we instead choose all the gates in the sequence independently (i.i.d) at random, with each block having the same width/duration. So every gate is of the form

$\exp(i \tau H_j )$

where the width $\tau$ is a constant. The coefficients in the Hamiltonian instead determining the probability (and hence frequency) with which a particular gate appears. The resulting sequence for our example Hamiltonian might look something like this:

The colour refers to which $H_j$ is used in the gate, using the same convention from earlier. Notice that the blue gates (for $H_1$) are the most frequent, comprising roughly half the gates in the whole sequence. The probabilities are tuned so that (on average) the same total width of any one colour is the same here as for Trotter above. In this illustration, we have 25 gates in total, half of the 50 for standard Trotter. The basic is idea is that such a random sequence can achieve the same simulation precision, having used fewer gates.

There are two effects at work here. First, we see that in the original Trotter there were a lot of weak gates (yellow, green and purple) that were repeated in every step. One of the main savings is that these are consolidated. Second, the sequence is random and so any coherent noise effects are washed out into less harmful stochastic noise. This is a very subtle point that I’ve blogged on previously, so I won’t labour the point again other than to stress that the protocol only works if a different random sequence is used for every run of the quantum computer.

Remarkably, one finds the number of gates required is independent of the number of terms in the Hamiltonian! This is proved by Taylor expanding and careful bounding of errors, much like the proofs for standard Trotter. One difference is that instead of Taylor expanding a unitary we perform a Taylor expansion of an exponentiated Lindblad operator.

For technical details, you should see the paper. To wrap up here, I just want to add that the above random protocol (which I call qDRIFT) is very simple and there is plently of scope for more sophisticated ideas exploiting the same effects. Right away one can see that there is some waste in the above gate sequence; whenever you see subsequent gates of the same colour, they can be merged into a single gate. Applying such a merge strategy to the above, one would have the sequence:

which has only 16 gates!

# Are quantum computers more powerful than classical computers and the stabiliser rank story. (part 1)

Of course, quantum computers are more powerful than classical computers, but how do you prove it! Until recently there was not much known except for some oracle separation results. In 2017, we got the already infamous Bravyi-Gosset-Koenig result that proved quantum computers can solve some problems in constant time that for a classical computer require logarithmic time. But we really want to know that quantum computers offer an exponential speedup such as Shor’s algorithm suggests.

Asking for a proof of an exponential speedup is a lot to ask for! We would need to show that all possible classical algorithms are incapable of efficiently simulating a quantum computer. This is really hard. But if we have a concrete family of classical algorithms, then surely we should try to understand if they could possibly offer an efficient simulation!

This brings us to stabiliser rank and related simulation methods. Let’s define some terms. Given a state $\Psi$ we define the stabiliser rank $\chi(\Psi)$ to be the smallest number of terms needed to write $\Psi$ as a sum of stabiliser states. More formally,

Definition of stabiliser rank:
Suppose $\psi$ is a pure $n$-qubit state. The exact stabilizer rank $\chi(\psi)$ is the smallest integer $k$ such that $\psi$ can be written as

$\vert \psi \rangle = \sum_{\alpha=1}^{k} c_\alpha \vert \phi_\alpha \rangle$

for some $n$-qubit stabilizer states $\phi_\alpha$ and some complex coefficients $c_\alpha$.

These decompositions were first studied by Garcia- Markov-Cross and applications to simulation were first explored in more depth by Bravyi-Smith-Smolin. The Gottesman-Knill theorem allows us to efficiently simulate a Clifford circuit acting on a stabiliser state. This can be extended to simulation of an arbitrary state $\Psi$ undergoing a Clifford circuit with runtime proportional to $\chi(\Psi)$ (assuming we know the decomposition). We also know that a universal quantum computation can be gadgetized into a Clifford circuit with a supply of single qubit magic states $\Psi = \psi^{\otimes t}$ where $t$ scales polynomially with the number of gates in the original quantum computation.

So if quantum computers are more powerful than classical computers, then we expect $\chi( \psi^{\otimes t})$ to scale exponentially with $t$. If $\chi( \psi^{\otimes t})$ instead scales polynomially with $t$ then BQP=BPP (you can classically efficiently simulate quantum computers). But also the classical simulation method allows for postselection, so it would actually entail postBQP=BPP, which entails NP=P.

Surely, this is a simple statement that the quantum community should be able to prove. Surely, stabiliser rank simulators are not going to show BQP=BPP. Surely, surely we can prove the following conjecture:

The stabiliser rank conjecture:
For every nonstabiliser state $\psi$, there exists a $\alpha >0$ such that

$e^{\alpha t} \leq \chi( \psi^{\otimes t} )$

holds for all integer $t$.

Clearly, one has that $\chi( \psi^{\otimes t}) \leq \chi( \psi)^t$, but the problem is that $\chi( \psi^{\otimes t}) \ll \chi( \psi)^t$ and that we have almost no lower bounds on the stabiliser rank. Consider, a 1-qubit non-stabiliser state

$\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle$

But when we consider 2 copies of the state we have

$\vert \psi^{\otimes 2} \rangle = \alpha^2 \vert 0 0 \rangle + \alpha \beta \sqrt{2} \vert \Psi^+ \rangle + \beta^2 \vert 1 1 \rangle$

where

$\vert \Psi^+ \rangle = ( \vert 0 1 \rangle + \vert 1 0 \rangle ) / \sqrt{2}$

This is a decomposition with 3 stabiliser terms and so

$\vert \psi^{\otimes 2} \rangle \leq 3$

for all single qubit $\psi$. Ok, what about for more copies? Amongst many results in our recent Bravyi-Browne-Calpin-Campbell-Gosset-Howard (herein BBCCGH) is a proof that

$\vert \psi^{\otimes t} \rangle \leq t+1$ for $t \leq 5$

(see Sec V.A of arXiv_V1). The stabiliser rank grows linearly! Not exponentially, but linearly! OK, this scaling holds up until only five copies, but I find this remarkable. Incidentally, the proof has a nice connection to the fact that the Clifford group forms a 3-design. Beyond five copies, the numerics look exponential, but the numerics are heuristic, inconclusive and only go slightly further than five copies!

Do we have any lower bounds on stabiliser rank! Deep in Bravyi-Smith-Smolin (Appendix C) is tucked away a neat result showing that for the $\vert T \rangle$ state (an especially important single-qubit state) we know the stabiliser rank must scale at least as fast as $\Omega( \sqrt{t} )$ for t copies. We are hoping for an exponential lower bound and yet the best we find is a square root. Not even a linear lower bound! I find this shocking. And I assure you, it is not for lack of many smart people spending many, many hours trying to get a better lower bound. In Sec V.D. of BBCCGH, we (I say “we” but this part was all Sergey) show an exponential lower bound but with an additional (and quite strong) constraint on the allowed stabiliser states in the decomposition. The proof technique is really interesting and hints at routes forward to solving the full problem.

There are also a whole bunch of cool open problems revolving around the approximate stabiliser rank (introduced by Bravyi-Gosset) and which BBCCGH show is connected to a quantity called “the extent” (see Def 3 of BBCCGH). I want to say more about these open problems but this post has already gone on too long. More to follow in part 2!

# Joint PhD position between Sheffield and Oxford

Circuit compilers for near-term quantum computers

Summary: The number of elementary gates in a quantum computation determines the runtime of the quantum computer. It is clearly advantageous to have faster computations that use fewer gates and “circuit compilation” is the art of optimizing and automating this process. For near-term quantum computers (without error correction) effective compilation is especially important because these devices will be noisy and this imposes a practical limit on the number of gates before an error becomes inevitable. Therefore, compilation protocols and software are crucial to whether we will be able to demonstrate a quantum advantage before full-blown error-corrected devices are available. This PhD project will develop compilation methods exploring random compilers and simulation of fermionic and bosonic interacting systems. This is a joint project between the University of Sheffield and Oxford University.

The studentship is fully funded for UK students with funding provided by NQIT, the UK national hub for quantum computing and Networked Quantum Information Technologies. As such, the student will have the opportunity to collaborate and contribute towards the UK’s largest quantum computing effort.

The project is a joint collaboration between the groups of Simon Benjamin in Oxford, and Earl Campbell in Sheffield. The project and student could be based at either university, according to the preference of the successful candidate. The studentship will be held at the chosen university and the student will be registered for a doctoral degree of the chosen university.

We are looking for an enthusiastic student with a physics, mathematics or computer science degree. The award should be a minimum of a UK upper second class honours degree (2:1). The student should also have some of the following desirable skills: a good undergraduate-level understanding of quantum mechanics, a strong mathematical background and/or experience programming and running numerical simulations (e.g. in C).

Informal inquiries can be made to: e.campbell@sheffield.ac.uk or simon.benjamin@materials.ox.ac.uk)

# QEC ’19 call for submissions.

Hi folks,

Here is a message from Dan about QEC ’19. It is coming to the UK!!! Very excited and pleased to be helping out with chairing the programme.

***

Dear colleagues,

I’m pleased to announce that the 5th International Conference on Quantum Error Correction (QEC’19) is now open for the submission of contributed talks and posters.

QEC ’19 will be taking place at Senate House, London between 29th July and 2nd August 2019. It aims to bring together experts from academia and industry to discuss theoretical, experimental and technological research towards robust quantum computation. Topics will include control, error correction and fault tolerance, and their interface with physics, computer science and technology research.

Please see our web-page for further information about the conference, including the list of invited speakers, and information on how to submit your one-page abstract: http://qec19.iopconfs.org/

The deadline for submissions is April 1st.

The scope of QEC is broad. Here is a non-exhaustive list of topics covered by the conference: Theoretical and experimental research in: Quantum error correcting codes; error correction and decoders; architecture and design of fault tolerant quantum computers; error mitigation in NISQ computations, error mitigation in quantum annealing; reliable quantum control, dynamical decoupling; Error quantification, tomography and benchmarking; Quantum repeaters; Active or self-correcting quantum memory. Applications of quantum error correction in Physics, CS and other disciplines; Classical error correction relevant to the QEC community; Compilers for fault tolerant gate sets; Error correction and mitigation in other Quantum Technologies (sensing / cryptography / etc.); QEC-inspired quantum computing theory.

Key dates

Abstract submission deadline: 1 April 2019
Registration opens: TBC (late February)
Early registration deadline: 10 June 2019

A conference poster is available at the following link:
https://www.dropbox.com/s/tpz4v0q406kq1e0/QEC%202019%20London.pdf?dl=0

I would be grateful if you would print this poster and display it in a place where interested participants may see it.

I look forward to your submission to QEC’19 and to seeing you in London in the summer.

Regards,

Dan Browne – Conference Chair

# The folly of fidelity and how I learned to love randomness

If a quantum device would ideally prepare a state $\psi$, but instead we prepare some other approximation state $\rho$. Then how should the error be quantified?

For many people, the gold standard is to use the fidelity. This is defined as

$F = \langle \psi \vert \rho \vert \psi \rangle$

One of the main goals of this blog post is to explain why the fidelity is a terrible, terrible error metric. You should be using the “trace norm distance”. I will not appeal to mathematical beauty — though trace norm wins on these grounds also — but simply to what is experimentally and observationally meaningful. What do I mean by this? We never observe quantum states. We observe measurement results. Ultimately, we care that the observed statistics of experiments is a good approximation of the exact, target statistics. In particular, every measurement outcome is governed by a projector $\Pi$ and if we have two quantum states $\psi$ and $\rho$ then the difference in outcome probabilities is

$\epsilon_\Pi = | \mathrm{Tr} [ \Pi \rho ] - \langle \psi \vert \Pi \vert \psi \rangle |$

Error in outcome probabilities is the only physically meaningful kind of error. Anything else is an abstraction. The above error depends on the measurement we perform, so to get a single number we need to consider either the average or maximum over all possible projectors.

The second goal of this post is to convince you that randomness is really, super useful! This will link into the fidelity vs trace norm story. Several of my papers use randomness in a subtle way to outperform deterministic protocols (see here, here and most recently here). I’ve also written another blog posts on the subject (here). Nevertheless, I still find many people are wary of randomness. Consider a protocol for approximating $\psi$ that is probabilistic and prepares states $\phi_k$ with probability $p_k$. What we will see is that the error (when correctly quantified) of probabilistic protocols can be considerably less than any of the pure states $\phi_k$ in this ensemble. Strange but true. Remember, I only care about measurement probabilities. And you should only care about measurement probabilities! Given a random protocol, the probability of a certain outcome will be
$\sum_k p_k \langle \phi_k \vert \Pi \vert \phi_k \rangle = \mathrm{Tr}[ \Pi \rho ]$
where
$\rho = \sum_k p_k \vert \phi_k \rangle \langle \phi_k \vert$
so our measure of error must depend on the averaged density matrix. Even though we might know after the event which pure state we prepared, the measurement statistics are entirely governed by the density matrix that averages over the ensemble of pure states. When I toss a coin, I know after the event with probability 100% whether it is heads or tails, but this does not prevent it from being an excellent 50/50 (pseudo) random number generator. Say I want to build a random number generator with a 25/75 split of different outcomes. I could toss a coin once: if I get heads then I output heads; otherwise I toss a second time and output the second toss result. This clearly does the job. We do not hear people object, “ah ha but after the first toss, it is now more likely to give tails so you algorithm is broken”. Similarly, quantum computers are random number generators and are allowed a “first coin toss” that determines what they do next.

Let’s thinking about the simplest possible example; a single-qubit. Imagine the ideal, or target state, is $\vert \psi \rangle = \vert 0 \rangle$. We consider states of the form $\vert \theta \rangle = \cos( \theta / 2 ) \vert 0 \rangle + \sin( \theta / 2 ) \vert 1 \rangle$. The states $\vert \theta \rangle$ and $\vert -\theta \rangle$ both have the same fidelity with respect to the target state. So too does the mixed state
$\rho(\theta) = \frac{1}{2} ( \vert \theta \rangle \langle \theta \vert + \vert -\theta \rangle \langle -\theta \vert )$.
Performing a measurement in the X-Z plane, we get the following measurement statistics.

We have three different lines for three states that all have the same fidelity, but the measurements errors behave very differently. Two things jump out.

Firstly, the fidelity is not a reliable measure of measurement error. For the pure states, in the worst case, the measurement error is 20 times higher than the error as quantified by fidelity. Actually, for almost all measurement angles the fidelity is considerably less than the measurement error for the pure states. Just about the only thing fidelity tells you is that there is one measurement (e.g. with $\varphi=0$) such that the fidelity tells you the measurement error. But it is not generally the case that the quantum computer will perform precisely this measurement. This closes the case on fidelity.

Secondly, the plot highlights how the mixed state performs considerably better than the pure states with the same fidelity! Clearly, we should choose the preparation of the mixed state over either of the pure states. For each pure state, there are very small windows where the measurement error is less than for the mixed state. But when you perform a quantum computation you will not have prior access to a plot like this to help you decide what states to prepare. Rather, you need to design your computation to provide a promise that if your error is $\delta$ (correctly quantified) then all probabilities should be correct up-to and not exceeding $\delta$. The mixed state gives the best promise of this kind.

Now we can see that the “right” metric for measuring errors should be something like:
$\delta = \mathrm{max}_{\Pi} | \mathrm{Tr} [ \Pi \rho ] - \langle \psi \vert \Pi \vert \psi \rangle |$
Nothing could be more grounded in experiments. This is precisely the trace norm (also called the 1-norm) measure of error. Though it needs a bit of manipulation to gets into it’s most recognisable form. One might initially worry that the optimisation over projections is tricky to compute, but it is not! We rearrange as follows

$\delta = \mathrm{max}_{\Pi} | \mathrm{Tr} [ \Pi (\rho- \vert \psi \rangle\langle \psi \vert) ] |$

let

$M = \rho- \vert \psi \rangle\langle \psi \vert = \sum_j \lambda_j \vert \Phi_j \rangle\langle \Phi_j \vert$

where we have given the eigenvalue/vector decomposition of this operator. The eigenvalues will be real numbers because $M$ is Hermitian.

It takes a little bit of thinking to realise that the maximum is achieved by a projection onto the subspace with positive eigenvalues
$\delta = \mathrm{max}_{\Pi} | \mathrm{Tr} [ \Pi M ] = \sum_{j : \lambda_j > 0} \lambda_j$
Because, $M$ is traceless, the eigenvalues sum to zero and we can simplify this as follows
$\delta = \frac{1}{2}\sum_{j } | \lambda_j |$
If we calculated this $\delta$ quantity and added it to the plots above, we would indeed see that it gives a strict upper bound on the measurement error for all measurements.

We have just been talking about measurement errors, but have actually derived the trace norm error. Let us connect this to the usual mathematical way of introducing the trace norm. For an operator $M$, we denote $|| M ||_1$ or $|| M ||_{\mathrm{tr}}$ for the trace norm, which simply means take the absoluate sum of the eigenvalues of $M$. Given two states, the trace norm distance is defined as
$\frac{1}{2} || \rho -\sigma ||_{\mathrm{tr}}$
Hopefully, the reader can see that this is exactly what we have found above. In other words, we have the equivalence
$\frac{1}{2} || \rho -\sigma ||_{\mathrm{tr}} = \mathrm{max}_{\Pi} | \mathrm{Tr} [ \Pi ( \rho -\sigma) ]$

Note that we need the density matrix to evaluate this error metric. If we have a simulation that works with pure states, and we perform statistic sampling over some probability distribution to find the average trace norm error
$\sum_{k} p_k \frac{1}{2} \big|\big| \vert \phi_k \rangle \langle \phi_k \vert - \vert \psi \rangle \langle \psi \vert \big|\big|_{\mathrm{tr}}$
then this will massively over estimated the true error of the density matrix
$\rho = \sum_{k} p_k \vert \phi_k \rangle \langle \phi_k \vert$

I hope this convinces you all to use the trace norm in future error quantification.

I want to finish by tying this back into my recent preprint:
arXiv1811.08017
That preprint looks not at state preparation but the synthesis of a unitary. There I make use of the diamond norm distance for quantum channels. The reason for using this is essentially the same as above. If we have a channel that has error $\delta$ in diamond norm distance, it ensures that for any state preparation routine the final state will have no more than error $\delta$ in trace norm distance. And therefore, it correctly upper bounds the error in measurement statistics.

# Explainer for topological quantum error correction

Explain topological quantum error correction in 2 minutes without too much jargon. Sure, we’ll give it a go: