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Pauli error estimation via Population Recovery



Steven T. Flammia1,2 and Ryan O’Donnell3

1AWS Center for Quantum Computing, USA
2IQIM, California Institute of Technology, USA
3Computer Science Department, Carnegie Mellon University, USA

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Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the “Population Recovery” problem, we give an extremely simple algorithm that learns the Pauli error rates of an $n$-qubit channel to precision $epsilon$ in $ell_infty$ using just $O(1/epsilon^2) log(n/epsilon)$ applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an $O(1/epsilon)$ factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability $le 1/4$.
We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability $1-eta$. In the regime of small $eta$ we extend our algorithm to achieve multiplicative precision $1 pm epsilon$ (i.e., additive precision $epsilon eta$) using just $Obigl(frac{1}{epsilon^2 eta}bigr) log(n/epsilon)$ applications of the channel.

The term “population recovery” is usually understood in the context of biology, where an endangered species (such as the gray wolf pictured here) is protected and their numbers begin to rebound. In the context of computer science, however, it refers to the ability to learn a probability distribution given only access to noisy samples. The “population” that we wish to learn (“recover”) is an unknown distribution on bit strings, and our ability to sample from this distribution is subject to independent noise, such as an erasure channel or a bit-flip channel.

In this work, we consider the problem of learning a probability distribution over Pauli operators; that is, we wish to learn a Pauli channel. Furthermore, we wish to do so using only very simple (product state) preparations and basis measurements. Learning Pauli channels with minimal resources is important for learning the errors in a noisy quantum computer and finding better ways to fix or otherwise mitigate those errors.

Our work shows that this problem reduces to the classical problem of population recovery with a certain type of asymmetric noise. We then show that the standard algorithms known in the literature for population recovery apply unchanged to this type of noise. This gives us very sample-efficient algorithms for learning Pauli channels that use very simple state preparations and measurements.

We also show a few other interesting tidbits. First, the algorithm is naturally robust to certain types of additional measurement noise. We can also extend the algorithm to handle the case where most of the time only a single Pauli occurs (say, the identity Pauli), and we wish to recover the remaining population with a precision that is relative to the frequency of the remainder population (a more stringent task). We show an interesting connection to Fourier analysis on boolean variables. Finally, we give an open source implementation in Julia of one of the algorithms.

► BibTeX data

► References

[1] F. Ban, X. Chen, A. Freilich, R. Servedio, and S. Sinha. Beyond trace reconstruction: Population recovery from the deletion channel. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science, pages 745–768, 2019, arXiv:1904.05532.

[2] F. Ban, X. Chen, R. A. Servedio, and S. Sinha. Efficient Average-Case Population Recovery in the Presence of Insertions and Deletions. In D. Achlioptas and L. A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/​RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 44:1–44:18, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, arXiv:1907.05964.

[3] L. Batman, R. Impagliazzo, C. Murray, and R. Paturi. Finding heavy hitters from lossy or noisy data. In Proceedings of the 16th Annual International Conference on Approximation Algorithms for Combinatorial Optimization Problems, pages 347–362, 2013.

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[6] C. Dankert. Efficient Simulation of Random Quantum States and Operators. PhD thesis, University of Waterloo, 2015, arXiv:quant-ph/​0512217.

[7] A. De, R. O’Donnell, and R. Servedio. Sharp bounds for population recovery. Theory of Computing, 16(6):1–20, 2020, arXiv:1703.01474.

[8] A. De, M. Saks, and S. Tang. Noisy population recovery in polynomial time. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science, pages 675–684, 2016, arXiv:1602.07616.

[9] Z. Dvir, A. Rao, A. Wigderson, and A. Yehudayoff. Restriction access. In Proceedings of the 3nd Annual Innovations in Theoretical Computer Science, pages 19–33, 2012.

[10] S. Flammia and J. Wallman. Efficient estimation of Pauli channels. ACM Transactions on Quantum Computing, 1(1):1–32, 2020, arXiv:1907.12976.

[11] S. T. Flammia. PauliPopRec, Github repository, 2021.

[12] A. Fujiwara and H. Imai. Quantum parameter estimation of a generalized Pauli channel. Journal of Physics A: Mathematical and General, 36(29):8093–8103, jul 2003.

[13] O. Goldreich and L. Levin. A hard-core predicate for all one-way functions. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 25–32, 1989.

[14] R. Harper, S. T. Flammia, and J. J. Wallman. Efficient learning of quantum noise. Nature Physics, 16(12):1184–1188, Aug 2020, arXiv:1907.13022.

[15] R. Harper, W. Yu, and S. T. Flammia. Fast estimation of sparse quantum noise. PRX Quantum, 2(1):010322, Feb 2021, arXiv:2007.07901.

[16] M. Hayashi. Quantum Information Theory. Springer Berlin Heidelberg, 2nd edition, 2017.

[17] J. Helsen, X. Xue, L. M. K. Vandersypen, and S. Wehner. A new class of efficient randomized benchmarking protocols. npj Quantum Information, 5(1):71, Aug. 2019, arXiv:1806.02048.

[18] E. Knill. Quantum computing with realistically noisy devices. Nature, 434(7029):39–44, mar 2005, arXiv:quant-ph/​0410199.

[19] S. Lovett and J. Zhang. Improved noisy population recovery, and reverse Bonami–Beckner inequality for sparse functions. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 137–142, 2015.

[20] S. Lovett and J. Zhang. Noisy population recovery from unknown noise. In Conference on Learning Theory, pages 1417–1431, 2017.

[21] A. Moitra and M. Saks. A polynomial time algorithm for lossy population recovery. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pages 110–116, 2013, arXiv:1302.1515.

[22] A. Montanaro and T. J. Osborne. Quantum Boolean functions. Chicago Journal of Theoretical Computer Science, 2010(1):1–45, January 2010, arXiv:0810.2435.

[23] S. Narayanan. Improved algorithms for population recovery from the deletion channel. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1259–1278. Society for Industrial and Applied Mathematics, Jan. 2021, arXiv:2004.06828.

[24] R. O’Donnell. Analysis of Boolean functions. Cambridge University Press, 2014, arXiv:2105.10386.

[25] Y. Polyanskiy, A. T. Suresh, and Y. Wu. Sample complexity of population recovery. In S. Kale and O. Shamir, editors, Proceedings of the 2017 Conference on Learning Theory, volume 65 of Proceedings of Machine Learning Research, pages 1589–1618, Amsterdam, Netherlands, 07–10 Jul 2017. PMLR, arXiv:1702.05574.

[26] B. Terhal. Quantum error correction for quantum memories. Reviews of Modern Physics, 87(2):307, 2015, arXiv:1302.3428.

[27] J. ur Rehman and H. Shin. Entanglement-free parameter estimation of generalized Pauli channels. Quantum, 5:490, Jul 2021, arXiv:2102.00740.

[28] M. Wainwright. High-dimensional statistics: A non-asymptotic viewpoint. Cambridge University Press, 2019.

[29] J. Wallman and J. Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Physical Review A, 94(5):052325, 2016, arXiv:1512.01098.

[30] A. Wigderson and A. Yehudayoff. Population recovery and partial identification. Machine Learning, 102(1):29–56, 2016.

Cited by

[1] Yunchao Liu, Matthew Otten, Roozbeh Bassirianjahromi, Liang Jiang, and Bill Fefferman, “Benchmarking near-term quantum computers via random circuit sampling”, arXiv:2105.05232.

[2] Thomas Wagner, Hermann Kampermann, Dagmar Bruß, and Martin Kliesch, “Pauli channels can be estimated from syndrome measurements in quantum error correction”, arXiv:2107.14252.

[3] Senrui Chen, Sisi Zhou, Alireza Seif, and Liang Jiang, “Quantum advantages for Pauli channel estimation”, arXiv:2108.08488.

[4] Steven T. Flammia, “Averaged circuit eigenvalue sampling”, arXiv:2108.05803.

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-23 14:17:36). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-09-23 14:17:34: Could not fetch cited-by data for 10.22331/q-2021-09-23-549 from Crossref. This is normal if the DOI was registered recently.

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The Simple Math Behind the Mighty Roots of Unity



If you’ve ever taken an algebra or physics class, then you’ve met a parabola, the simple curve that can model how a ball flies through the air. The most important part of a parabola is the vertex — its highest or lowest point — and there are many mathematical techniques for finding it. You can try vertex form, or the axis of symmetry, or even calculus.

But last week one of my students located the vertex of a parabola in a particularly elegant way. “The vertex is at x = 4,” she said, “because the roots are x = 1 and x = 7, and the roots are symmetric about the vertex.” She used the fact that the parabola is the graph of a quadratic polynomial, and that the roots of that polynomial — the values where it becomes 0 — have a certain structure she could take advantage of.

There is a structure to the roots of every polynomial, and mathematicians study these structures and look for opportunities to capitalize on them, just as my student did with her parabola. And when it comes to the roots of polynomials, none have more structure than the “roots of unity.”

Roots of unity are the roots of the polynomials of the form xn – 1. For example, when n = 2, this gives us the quadratic polynomial x2 – 1. To find its roots, just set it equal to 0 and solve:

 x2 – 1 = 0.

You might remember factoring expressions like this using the “difference of squares” formula, which says that a2 – b2 = (ab)(a + b). Here x2 – 12 = (x – 1)(x + 1), which gives you

(x – 1)(x + 1) = 0.

Now that you’ve got a product equal to 0, you can invoke one of the most underappreciated rules from algebra class: the “zero product property.” This says that the only way two real numbers can multiply to 0 is for one of them to be 0. So if (x – 1)(x + 1) = 0, then either x – 1 = 0 or x + 1  = 0. The first equation is true when x = 1, the second when x = −1. So 1 and −1 are the two “second roots of unity,” which might be more familiar to you as the two square roots of 1.

For any n you can find the nth roots of unity, which are the solutions to the equation xn – 1 = 0. These roots of unity possess a remarkably rich structure that connects to high school math like trigonometry and rotations of the plane as well as ongoing research that involves some of the great unanswered questions in modern math.

When n = 2, the two roots 1 and −1 have a symmetric structure that is related to how my student found her vertex. You can see even more structure in the fourth roots of unity. These are the solutions to the equation x= 1. You might recognize two of the fourth roots of unity right away: Since 1= 1 and (−1)= 1, x = 1 and x = −1 both satisfy the equation, so they are fourth roots of unity. But there are actually two more, and you can find them using algebra as we did above: Just put the equation into standard form and factor:

x= 1
x– 1 = 0

Since xand 1 are both perfect squares, you can use the difference of squares formula here as well:

x– 1 = (x2)– 1= (x– 1)(x+ 1). This turns the equation x– 1 = 0 into

(x– 1)(x+ 1)=0.

The x– 1 should look familiar: We factored that when we found the square roots of unity. This gives us

(x – 1)(x + 1)(x+ 1) = 0.

We can’t factor any more right now. The expression x+ 1 is “irreducible” over the real numbers, which means it can’t be broken down into simpler multiplicative factors that involve only real numbers. But we can still apply the zero product property. If these three numbers multiply to 0, then one of them must be zero. That is, either x – 1 = 0, x + 1 = 0, or  x+ 1 = 0.

The first two equations tell us what we already knew: x = 1 and x = −1 are solutions to the equation x= 1 and are therefore fourth roots of unity. But what can we do with x+ 1 = 0?

Well, if you know about complex numbers, then you know that i, the “imaginary unit,” satisfies this equation because it is defined by the property that i= −1. It isn’t a real number — no real number squared is negative — but it turns out that most roots of unity are complex numbers, and since satisfies x+ 1 = 0, it must be one of the fourth roots of unity. You can easily verify this with some rules of exponents: Since i= −1, then i= (i2)= (−1)= 1. And since complex numbers follow most rules that real numbers follow, it’s true that (−i)i, and so x = −i  also satisfies x+ 1 = 0 and is also a fourth root of unity.

These four numbers, 1, −1, i, and −i, are all fourth roots of unity, and the matching fours are not a coincidence. The fundamental theorem of algebra says that every nth-degree polynomial has n complex roots. Therefore the equation x= 1 has n complex solutions, and these are all the nth roots of unity. (Since real numbers are also complex numbers, the real solutions, like 1 and −1, are included in the count of complex solutions.)

For a given n the nth roots of unity possess some remarkable properties. Geometrically, if you graph the nth roots of unity in the complex plane you’ll find that they are equally spaced around the unit circle centered at the origin.

This geometric structure is closely connected to important ideas in trigonometry, like the angle sum and difference formulas for sine and cosine, the theory of rotations of the plane, and e, the base of the natural logarithm function. This geometry is also connected to an interesting algebraic property: For any n, the sum of the nth roots of unity is 0.

For n = 2 this is immediately obvious: The sum of both square roots of unity is 1 + (−1) = 0. It’s clear, too, for the four fourth roots of unity:

1 + + (−1) + (−i) = 0.

In both cases it’s easy to see why the sum is 0: The roots of unity come in opposite pairs, which cancel out when you add them up.

However, the result holds even when the roots of unity don’t come in opposite pairs. For example, the three third roots of unity are 1, $latex -frac{1}{2}+i frac{sqrt{3}}{2}$, and $latex -frac{1}{2} – i frac{sqrt{3}}{2}$. The two non-real roots don’t cancel out, but they do sum to −1, which then cancels out with the remaining root of unity, giving you 0 in the end:

1 + $latex left(-frac{1}{2}+i frac{sqrt{3}}{2}right)+left(-frac{1}{2}-i frac{sqrt{3}}{2}right)$ = 1 + (−1) = 0.

You can establish this property geometrically, but there’s an elegant algebraic argument that shows this is true. Let’s call the three third roots of unity 1, α and β. All three of these numbers satisfy the cubic equation

x– 1 = 0.

Because you know the roots of this cubic equation, you know that the polynomial on the left factors as

(x – 1)(x – α)(x – β) = 0.

If you multiply this expression out using the distributive property a few times, you get the following:

x3 – (1 + α + β)x2 + (α + β + αβ)x – αβ = 0.

But we already know what cubic polynomial we should get when we multiply this out: x3 – 1. So x3 – (1 + α + β)x2 + (α + β + αβ)x – αβ is really x3 – 1, which means that the coefficient of x2 on the left side, 1 + α + β, has to equal the coefficient of x2 on the right side, which is 0. Thus 1 + α + β =0, and so the three third roots of unity sum to 0.

This argument generalizes and produces one of “Vieta’s formulas,” which are famous results that relate the roots of a polynomial to its coefficients. One of Vieta’s formulas says that, in a polynomial that begins with xn, the sum of the roots of the polynomial will always be the negation of the coefficient of xn-1. Since roots of unity come from polynomials of the form xn– 1, where the coefficient of xn-1 is always 0, Vieta’s formula tells us that the sum of the nth roots of unity is 0 for any n.

There’s an even more remarkable algebraic result when it comes to roots of unity. For a given n, if α and β are two nth roots of unity, then α × β is also an nth root of unity! And if α and β are both nth roots of unity, then α= 1 and β= 1. So what is (α × β)n?

In general you have to be careful raising complex numbers to a power, but since the n in the nth roots of unity is assumed to always be an integer, the basic rules of exponents still apply, like this one:

(α × β)= αn × βn .

So (α × β)= αn × β= 1 × 1 = 1. This means α × β satisfies the equation x= 1 and so is an nth root of unity. For example, when n = 4, if you multiply the two roots of unity i and −1, you get another fourth root of unity:  i × (−1) = −i. And when n = 3, you can also verify by multiplication that the two non-real roots of unity multiply to the real one: $latex left(-frac{1}{2}+i frac{sqrt{3}}{2}right)×left(-frac{1}{2}-i frac{sqrt{3}}{2}right)$ = 1.

This property gives rise to an incredibly rich algebraic structure on the nth roots of unity: a “group” structure. A group is a set of elements (here, the nth roots of unity) and an operation (here, normal multiplication) that satisfies some familiar properties. One of those properties is “closure,” which we just demonstrated. This means the product of two nth roots of unity is always another nth root of unity. Another important property of groups is that inverses always exist. This means that for any nth root of unity there is another nth root of unity such that their product is 1, the multiplicative identity. For example, when n = 4, the inverse of i is −since i × (−i) = −i2 = −(−1) = 1, and among the third roots of unity the inverse of  $latex -frac{1}{2}+i frac{sqrt{3}}{2}$ happens to be $latex -frac{1}{2}-i frac{sqrt{3}}{2}$.

The study of groups is fundamental to Galois theory, an advanced field of mathematics built to study abstract algebraic structures associated with polynomials and their roots. You probably know the quadratic formula and possibly know of the cubic and quartic formulas, but there is no general formula for finding the roots of a polynomial of degree 5 or higher, and Galois theory helps unravel this mystery by studying the groups associated with the roots of polynomials.

Because the nth roots of unity have their own group structure, they occupy an important place in Galois theory, especially because that structure is so easy to work with. Roots-of-unity groups are always “abelian,” meaning that the order in which you multiply objects doesn’t change the result, and they are always “cyclic,” meaning that you can always generate the entire group by multiplying a single element by itself over and over.

In Galois theory, being associated with an abelian group is a very nice property for a polynomial, and the impact of the roots of unity extend well beyond just polynomials of the form xn − 1. It turns out that any polynomial associated with an abelian group in Galois theory has roots that can be expressed as sums of different roots of unity. In a sense, the roots of unity form the foundation of all the nice polynomials in a particular mathematical world, and generalizing the role of roots of unity to other mathematical worlds has been the goal of Hilbert’s 12th problem, one of the 23 math problems posed by David Hilbert in 1900 to guide the course of mathematical discovery for the next 100 years. Now, more than a century later, people are still working on the 12th problem, and progress is being made, but mathematicians aren’t all the way there yet. Perhaps soon they’ll get to the root of it all.


1. Show that the four fourth roots of unity are also eighth roots of unity.

2. Find the other four eighth roots of unity. (Hint: $latex sqrt{i}$ is one of them, but you need to write it in a + bi form.)

3. When is an nth root of unity also an mth root of unity?

4. A “primitive nth root of unity” is an nth root of unity whose powers include all the nth roots of unity. For example, i is a primitive fourth root of unity, since the powers of i are i, -1, -i, and 1, all four of the fourth roots of unity. But −1 is not a primitive fourth root of unity, since the powers of −1 are just −1 and 1.

Which of the eighth roots of unity are primitive?

Challenge: What is the product of all n of the nth roots of unity? (Hint: Take a closer look at Vieta’s formulas.)


Click for Answer 1:

Let α be a fourth root of unity. Then α4 = 1. Now square both sides of the equation to get α8 = 1. Since α satisfies x8 = 1, it is an eighth root of unity.

Click for Answer 2:

Notice that $latex sqrt{i}$ is an eighth root of unity, since if α = $latex sqrt{i}$, then α2 = i, and so α8 = i4 = 1. To find $latex sqrt{i}$, set $latex sqrt{i}$ = a + bi and square both sides to get i = a2b2 + 2abi. This tells you that a2b2 = 0 and 2ab = 1, and some algebra establishes that one square root of i is $latex frac{sqrt{2}}{2}$ + $latex i frac{sqrt{2}}{2}$. The other three eighth roots of unity are $latex frac{sqrt{2}}{2}$ – $latex i frac{sqrt{2}}{2}$, $latex −frac{sqrt{2}}{2}$ + $latex i frac{sqrt{2}}{2}$, and $latex −frac{sqrt{2}}{2}$ – $latex i frac{sqrt{2}}{2}$.

Click for Answer 3:

This is true when m is a multiple of n, and it follows from simple rules of exponents. Suppose αn = 1 and m = kn. Then αm = αkn = (αn)k = (1)k = 1, so α is an mth root of unity as well.

Click for Answer 4:

Let α = $latex frac{sqrt{2}}{2}$ + $latex i frac{sqrt{2}}{2}$. Then α is a primitive eighth root of unity, since α2 = i, α3 = $latex -frac{sqrt{2}}{2}+i frac{sqrt{2}}{2}$, α4 = -1, α5 = $latex -frac{sqrt{2}}{2}$ – $latex i frac{sqrt{2}}{2}$, α6 = –i, α7 = $latex frac{sqrt{2}}{2}$ – $latexi frac{sqrt{2}}{2}$, and α8 = 1. Some multiplication, or some clever use of laws of exponents, will show you that the other primitive eighth roots of unity are α3, α5 and α7. In fact, there’s a very beautiful relationship between n and the primitive nth roots of unity, which you can find just using laws of exponents. See if you can find it!

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Single Cells Evolve Large Multicellular Forms in Just Two Years



To human eyes, the dominant form of life on Earth is multicellular. These cathedrals of flesh, cellulose or chitin usually take shape by following a sophisticated, endlessly iterated program of development: A single microscopic cell divides, then divides again, and again and again, with each cell taking its place in the emerging tissues, until there is an elephant or a redwood where there was none before.

At least 20 times in life’s history — and possibly several times as often — single-celled organisms have made the leap to multicellularity, evolving to make forms larger than those of their ancestors. In a handful of those instances, multicellularity has gone into overdrive, producing the elaborate organisms known as plants, animals, fungi and some forms of algae. In these life forms, cells have shaped themselves into tissues with different functions — cells of the heart muscle and cells of the bloodstream, cells that hold up the stalk of a wheat plant, cells that photosynthesize. Some cells pass their genes on to the next generation, the germline cells like eggs and sperm, and then there are all the rest, the somatic cells that support the germline in its quest to propagate itself.

But compared to the highly successful simplicity of single-celled life, with its mantra of “eat, divide, repeat,” multicellularity seems convoluted and full of perilous commitments. Questions about what circumstances could have enticed organisms to take this fork in the road millions of years ago on Earth — not once but many times — therefore tantalize scientists from game theorists and paleontologists to biologists tending single-celled organisms in the lab.

Now, the biologist William Ratcliff at the Georgia Institute of Technology and his colleagues report that over the course of nearly two years of evolution, they have induced unicellular yeasts to grow into multicellular clusters of immense size, going from microscopic to branching structures visible to the naked eye. The findings illustrate how such a transition can happen, and they imply intriguing future experiments to see whether these structures develop differentiation — whether cells start to play specialized roles in the drama of life lived together.

Incentives to Be Snowflakes

Nearly a decade ago, scientists who study multicellularity were set abuzz by an experiment performed by Ratcliff, Michael Travisano, and their colleagues at the University of Minnesota. Ratcliff, who was doing his doctoral thesis on cooperation and symbiosis in yeasts, had been chatting with Travisano about multicellularity, and they wondered whether it might be possible to evolve yeast into something multicellular. On a whim, they took tubes of yeast growing in culture, shook them, and selected the ones that settled to the bottom fastest to seed a new culture, over and over again for 60 days.

This simple procedure, as they later described in the Proceedings of the National Academy of Sciences, rapidly caused the evolution of tiny clumps — yeasts that had evolved to stay attached to each other, the better to survive the selection pressure exerted by the scientists. The researchers subsequently determined that because of a single mutation in ACE2, a transcription factor, the cells did not break apart after they divided, which made them heavier and able to sink faster.

This change in the cells emerged quickly and repeatedly. In less than 30 transfers, one of the tubes exhibited this clumping; within 60 transfers, all of the tubes were doing it. The researchers dubbed the cells snowflake yeast, after the ramifying shapes they saw under the microscope.

Snowflake yeast started out as a side project, but it looked like a promising avenue to explore. “That’s been my life for 10 years since then,” Ratcliff said. The work garnered him collaborators like Eric Libby, a mathematical biologist at Umeå University in Sweden, and Matthew Herron, a research scientist at Georgia Tech, where Ratcliff is now a professor. He had joined the varied ecosystem of researchers trying to understand how multicellular life came about.

It’s easy for us, as the vast architectures of cells that we are, to take it for granted that multicellularity is an unqualified advantage. But as far as we can tell from fossils, life seems to have been cheerfully unicellular for its first billion years. And even today, there are far more unicellular organisms than multicellular ones on the planet. Staying together has serious downsides: A cell’s fate becomes tied to those of the cells around it, so if they die, it may die too. And if a cell does become part of a multicellular collective, it may end up as a somatic cell instead of a germ cell, meaning that it sacrifices the opportunity to pass on its genes directly through reproduction.

There are also questions of competition. “Cells of the same species tend to compete for resources,” said Guy Cooper, a theorist at the University of Oxford. “When you stick a bunch of them together, that competition for resources becomes even stronger. That’s a big cost … so you need a benefit that’s equal or greater on the far side for multicellularity to evolve.”

One incentive might be that larger groups of cells can be harder for predators to eat. Independent work by Roberta Fisher at VU University Amsterdam in 2015 and Stefania Kapsetaki at Oxford in 2019 showed that algae and bacteria responded to predatory protozoa by forming groups. Herron and his colleagues showed in 2019 that this adaptive multicellularity in algae did not depend on the reappearance of some buried ancestral trait: It was a fully original, evolved adaptation.

Another possible incentive for multicellularity could be that organisms move better or forage better as a group under certain conditions. If that’s the case, Cooper explained, “that leads to a viability-fecundity trade-off, in the sense that you increase your survival at the cost of being less reproductive, because you’re competing for the resources.”

Some algae can switch between multicellular groups and single cells when their environments change. Choanoflagellates, the closest single-celled relatives to animals, can also opt to take actions that make them look curiously multicellular. Thibaut Brunet, an evolutionary biologist at the Pasteur Institute, recalls a workshop in Curaçao where he and colleagues collected water near the shore to check for choanoflagellates and noticed late at night, after dinner, that there was something moving in their sample. It was a new species of choanoflagellate that had joined together to form a cup shape, which was flipping itself inside out to move. “It was mesmerizing to see this thing just deform. … It had this complex collective behavior that made it almost animal-like,” Brunet said. “You could almost feel that transition from the microbial world to the animal world.”

But for the cells of most multicellular creatures, there is no choice — it’s multicellularity or death. “It somehow becomes a one-way road,” said Cooper. “And division of labor is predicted to be a big player in that transition.” Once some cells start to perform a new role, giving up their own reproductive success to increase that of their neighbors, computational models suggest that living in a group must provide efficiency benefits for the lifestyle to stand a chance of surviving. The parameters required for success must have been met in the past, but how exactly?

When Ratcliff began his long-term experimental evolution work, he combined a theorist’s interest in myriad possible scenarios with a biologist’s curiosity about what a real, living organism would do when pressed to the limit. He was also thinking about one of the most famous evolution experiments, started by Richard Lenski more than 30 years ago: 12 E. coli colonies in Lenski’s lab have been maintained since 1988. They’ve morphed over the years in surprising ways: For instance, in 2003, Lenski and his colleagues found that one population had evolved the ability to digest citrate, which E. coli had never been known to do before.

Ratcliff wondered what would happen to snowflake yeast grown that long — would they eventually achieve large size? Would that lead to differentiation?

The snowflake yeast achieved multicellularity readily, but their clumps remained microscopic, no matter what Ratcliff tried. For years he failed to make progress, and he credits Ozan Bozdağ, a research scientist at Georgia Tech who was a postdoc in Ratcliff’s lab, with breaking through the wall.

Living Large Without Oxygen

The crucial ingredient turned out to be oxygen. Or rather, a lack of it.

Oxygen can be very helpful for living things, because cells can use it to break down sugars for massive energy payouts. When oxygen isn’t present, cells must ferment sugars instead, for a smaller usable yield. All along, Ratcliff had been growing yeast with oxygen. Bozdağ suggested growing some cultures without it.

Bozdağ began the selection experiments with three different groups of snowflake yeasts, two that could use oxygen and one that, because of a mutation, could not. Each group consisted of five genetically identical tubes, and Bozdağ mounted them in a shaking machine. Around the clock, the yeast were shaken at 225 revolutions per minute. Once a day, he let them settle on the counter for three minutes, then used the contents of the bottom of the tube to start fresh cultures. Then, back in the shaker they went. Every day in 2020 and early 2021, even during the lab closures of the COVID-19 pandemic, Bozdağ was there, with a special exemption granted by the university, exerting selection on the yeast.

During the first 100 days, the clusters in all 15 of the tubes doubled in size. Then they mostly plateaued until around the 250th day, when the sizes in two of the tubes that didn’t use oxygen started to creep upward again. Around day 350, Bozdağ noticed something in one of those tubes. There were clusters he could see with the naked eye. “As an evolutionary biologist … you think it’s a chance event. Somehow they got big, but they are going to lose out against the small ones in the long run — that is my thinking,” he said. “I didn’t really talk about this with Will at the time.”

But then clusters showed up in the second tube. And around day 400, the three other tubes of mutants that couldn’t use oxygen kicked into gear, and soon all five tubes had massive structures in them, topping out at about 20,000 times their initial size. Bozdağ started taking pictures of the clusters with his phone camera. There was no longer a need for a microscope.

Why did reliance on oxygen seem to cap the expansion of the yeast clusters? Oxygen diffuses through cells at a fixed rate, so as clusters get bigger, oxygen can reach the cells in the interior only slowly if at all. Although bigger clusters had a survival advantage within this experiment, the allure of oxygen was so compelling for yeasts that they limited the size of their clusters rather than forsake it. For the oxygen-independent mutants that relied on fermentation for energy, there was no disincentive to getting bigger.

But size wasn’t the only difference in the clusters. When the team looked at the big clusters under the microscope, it was clear the yeast had changed. The cells were more elongated, and while the first snowflake yeast clusters split apart easily — they had one-hundredth the cohesion of gelatin — the big clusters were much hardier. “They evolve from this really brittle material to something that has the material properties of wood,” said Ratcliff. “They get at least 10,000 times tougher.” The snowflake branches were tangled around each other, too, so that even when the shaking did break bonds, the pieces stayed together, enmeshed in the larger mass of their brethren. Biophysically, this suggests that a unicellular organism can evolve a way to maintain the physical integrity of a larger size.

That’s intriguing because large size and differentiation have been theorized to go hand in hand, Cooper explained. Fourteen years ago, the evolutionary biologist J.T. Bonner noted that the larger a multicellular organism is, the more cell types it generally has. He hypothesized that greater size demands an increase in complexity. The idea is that as organisms grow larger, they have a greater variety of needs to attend to. “This can provide an incentive to divide labor,” Cooper said, while noting that this may not always be the case.

You can see, then, how greater size could catalyze a change. Imagine a wad of snowflake yeast, growing larger and larger with each cell division. The outer branches are exposed to the nutrients and dangers of the outside world. The branches deep inside the cluster have a different experience; for them, nutrients are scarcer and the physical stresses may be greater. What if the cells inside began to behave differently from those on the outside? They might alter their metabolism to make do with less. They might grow sturdier cell walls to stand up to pressure, like the cells in the Ratcliff lab’s experiments. Or they might develop highly branched channels that funnel nutrients deeper into the cluster, a rudimentary circulatory system. Differences could creep into the behaviors and properties of cells in distant regions of a large cluster.

Imagine, then, that every time a new cluster forms, its experience recapitulates this process, with the same differences in the environments of the inner and outer cells driving the same divergent responses. You begin to see how the story of what was once a unicellular creature can be rewritten, its body a palimpsest of what it did to survive.

From Multicellularity to Differentiation

As yet, there are no documented cases of an organism evolving both multicellularity and regulated differentiation in the lab. The closest so far may be the snowflake yeast described in the 2012 paper of Ratcliff and his colleagues, in which cells at the juncture of two branches sometimes provoked their own deaths. That caused the branches attached to the dead cell to break off and start clusters of their own. The team believes this could be a form of differentiation, in that the cells giving up their lives may have benefited the yeast as a group. “There may be some benefit of cell death, if it breaks apart cells before they run into limited nutrients,” said Libby, who worked with Ratcliff on modeling the phenomenon.

But he also notes that work by Paul Rainey of the Max Planck Institute for Evolutionary Biology and his colleagues has shown that Pseudomonas bacteria can also form multicellular groups in which cells may take on different forms and behaviors that serve a collective end. Identifying true differentiation in these cases can be tricky. “Honestly, these statements can be debatable because primitive forms of multicellular complexity often look like typical unicellular behavior,” Libby said. “This is no coincidence; it has to evolve from somewhere.”

It’s still highly speculative whether future experiments will show that massive snowflake yeasts can develop sophisticated differences in their tissues. But as the team continues evolving the yeast, there could be a lot of opportunities for strange things to happen.

Bozdağ recalls that when he told Ratcliff that the yeast had evolved large size, Ratcliff said, “Dude! You have to keep this going for 20, 30 years!” After years of disappointment, Ratcliff was thrilled to see that the yeasts could, in fact, provide themselves with something like a body.

“I wasn’t honestly sure if this was a system that would saturate at 1,000 or so cells,” Ratcliff said. “We have to continue evolving them and see what they can do. We need to see, if we push these guys as far as we can for decades, for tens of thousands of generations …”

He trailed off, then started again. “If we don’t do that, I will always regret not having taken the opportunity. It’s a once-in-a-lifetime opportunity, to try to push a nascent multicellular critter to become more complex and see how far we can take them.”

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Quantum Error Mitigation using Symmetry Expansion



Zhenyu Cai

Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom
Quantum Motion Technologies Ltd, Nexus, Discovery Way, Leeds, LS2 3AA, United Kingdom

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Even with the recent rapid developments in quantum hardware, noise remains the biggest challenge for the practical applications of any near-term quantum devices. Full quantum error correction cannot be implemented in these devices due to their limited scale. Therefore instead of relying on engineered code symmetry, symmetry verification was developed which uses the inherent symmetry within the physical problem we try to solve. In this article, we develop a general framework named symmetry expansion which provides a wide spectrum of symmetry-based error mitigation schemes beyond symmetry verification, enabling us to achieve different balances between the estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 6 to 9 times below what is achievable by symmetry verification when the average number of circuit errors is between 1 to 2. The corresponding sampling cost for random shot noise reduction is just 2 to 6 times higher than symmetry verification. Beyond symmetries inherent to the physical problem, our formalism is also applicable to engineered symmetries. For example, the recent scheme for exponential error suppression using multiple noisy copies of the quantum device is just a special case of symmetry expansion using the permutation symmetry among the copies.

Mitigating errors using the symmetries of the problem is expected to play key roles in practical near-term quantum applications. A good example is symmetry verification, in which we perform measurements to verify whether the output state is in the right symmetry subspace, so that we can detect certain errors on the state. In this work, we introduce a general framework named symmetry expansion for symmetry-based error-mitigation techniques. It encompasses previous schemes like symmetry verification, symmetry subspace expansion and virtual distillation while also generates a wide range of schemes, offering different balances between the estimation biases and the sampling cost. Furthermore, we devise a way to identify the symmetry expansion schemes that have much smaller estimation biases than symmetry verification. These symmetry expansion schemes allow for cancellation between the biases due to the detectable and undetectable errors, while symmetry verification can only remove the biases due to the detectable errors, leaving behind the biases due to the undetectable errors.

► BibTeX data

► References

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Cited by

[1] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, “Error mitigation with Clifford quantum-circuit data”, arXiv:2005.10189.

[2] Bálint Koczor, “Exponential Error Suppression for Near-Term Quantum Devices”, arXiv:2011.05942.

[3] Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, “Qubit-efficient exponential suppression of errors”, arXiv:2102.06056.

[4] Ryan LaRose, Andrea Mari, Sarah Kaiser, Peter J. Karalekas, Andre A. Alves, Piotr Czarnik, Mohamed El Mandouh, Max H. Gordon, Yousef Hindy, Aaron Robertson, Purva Thakre, Nathan Shammah, and William J. Zeng, “Mitiq: A software package for error mitigation on noisy quantum computers”, arXiv:2009.04417.

[5] Nobuyuki Yoshioka, Hideaki Hakoshima, Yuichiro Matsuzaki, Yuuki Tokunaga, Yasunari Suzuki, and Suguru Endo, “Generalized quantum subspace expansion”, arXiv:2107.02611.

[6] Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, “Unifying and benchmarking state-of-the-art quantum error mitigation techniques”, arXiv:2107.13470.

[7] Zhenyu Cai, “Resource-efficient Purification-based Quantum Error Mitigation”, arXiv:2107.07279.

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-21 17:00:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-09-21 17:00:43: Could not fetch cited-by data for 10.22331/q-2021-09-21-548 from Crossref. This is normal if the DOI was registered recently.

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