Can one trust quantum simulators?

In the paper

“Recently there has also been substantial progress in investigations of other possible candidates for AQS, such as photonic systems [31, 32, 34-36], photonic and polariton systems [33, 38-42], artificial lattices in solid state systems [44].”

What the sources say

• Partly supported

  The Abstract (page unavailable) reports a photonic implementation of Shor’s algorithm, supporting this paper as an example of progress with photonic quantum systems. It describes quantum computation, however, not photonic systems as candidates for AQS; the other platform categories in the citing sentence have separate citations. (LLM, conf=0.84)

  cites [31] sourceExperimental — accuracy not yet measured
• Partly supported

  The paper reports a photonic, all-optical implementation of a quantum-computing algorithm (pp. 1–2), supporting its use as an example of progress with photonic systems. It does not identify that experiment as analog quantum simulation or establish photonic systems as candidates for AQS. (LLM, conf=0.83)

  cites [32] sourceExperimental — accuracy not yet measured
• Supported

  The paper describes an experimental photonic analogue: ultrashort laser pulses create an analogue event horizon, and the authors report spontaneous photon emission (p. 1). It therefore supports the citation's photonic-systems example of progress in investigating candidates for analogue quantum simulation; it is not evidence for the separately cited polariton or solid-state examples. (LLM, conf=0.87)

  cites [34] source
• Not checked yetcites [35] Kitagawa T Broome M Fedrizzi A Rudner M Berg E Kassal I Aspuru -Guzik A Demler E White A G, 2011
• Supported

  The citation is attached to “photonic systems,” not to the separate claims about polariton systems or solid-state lattices. In “Quantum monogamy and complementarity” (p. 6), the paper describes its all-optical analog quantum simulator and reports generating and characterizing photonic states, making it an example of progress in photonic quantum simulation. (LLM, conf=0.93)

  cites [36] source
• Not checked yetcites [33] source
• Supported

  The opening passage (p. 1) introduces an array of coupled optical cavities whose interacting photon–atom states are polaritons, and predicts a quantum phase transition. The Conclusions (p. 3) describe the system as a potential tunable quantum simulator. This supports citing the paper as an example of progress on photonic and polaritonic candidates; the artificial-lattice example has its own citation. (LLM, conf=0.91)

  cites [38] source
• Supported

  The citation is attached to the photonic-and-polariton-systems category, not the separate artificial-lattices category. The opening passage (p. 1) proposes polaritons in cavity arrays as a quantum simulator, and the Introduction (p. 1) describes the system as an alternative way to simulate many-body Hamiltonians. (LLM, conf=0.91)

  cites [39] source
• Supported

  The Introduction (p. 1) proposes coupled electromagnetic cavities as an engineered quantum many-body system with polaritonic excitations. The conclusion (p. 4) says the arrays could simulate XY spin models, supporting their identification as a photonic or polaritonic candidate for analog quantum simulation. This citation need not support the sentence’s separate example of artificial solid-state lattices. (LLM, conf=0.90)

  cites [40] source
• Not checked yetcites [41] source

In the paper

“We show evidence that QSs appear to work better in regimes that are classically easier to solve or simulate (high-temperature states), thus hinting at a connection between the amount of quantum correlations and the robustness of a QS.”

In the paper

“In Section VII, we show that the ground-state expectation values of certain local observables appear fairly robust under disorder, while this need not be true for the global many-body state of the simulator.”

In the paper

“A: The correlation length ξ decreases with disorder r , and its peak broadens (shown for a chain of 400 spins).”

In the paper

“As seen in Fig. 1B, the reduced simulator fidelities are much more robust to disorder than the global one - near the phase transition, f 2 ( r, λ ) decreases to approximately 0.998, and f 1 ( r, λ ) remains above 0.999.”

In the paper

“As Fig. 1A shows, however, disorder suppresses correlations and broadens the peak of ξ , thus making an extraction of the critical point much less reliable.”

In the paper

“C: The central charge c (full circles, left axis), extracted from a fit to the part-chain entropy, and the critical exponent ν (open circles, right axis), extracted from a collapse of the correlations in different chain lengths. Both change with disorder, which can lead to erroneously assigning the QPT to an incorrect universality class. However, the change begins relatively smoothly at low levels”

In the paper

“The AQS keeps a high fidelity in the case of a local quench, while it performs poorly for the global quench, with fidelities reaching lows of 0 . 8 even for small systems of 50 spins.”

In the paper

“Disorder reduces the correlation length of the spin chain.”

In the paper

“Again, for low temperatures fidelity worsens, but saturates to a few percent. For high temperatures, it is simple to perform an expansion of the fidelity which shows that f 1 ≃ 1 -T -2 .”

In the paper

“As with the classical algorithms [192], the AQS remains faithful when the state is almost classical (high temperatures). The simulator fidelity decreases rapidly for low temperatures, although it saturates at a fairly high value.”

What the sources say

• Not checked yetcites [192] source

In the paper

“Increasing disorder suppresses this, indicating the decrease of entanglement in the system (see Fig. 1C of the main text).”

In the paper

“Moreover, the crossing point becomes less well defined with increasing disorder showing that this analysis should be corrected for the presence of disorder.”

In the paper

“Finally, at large disorder, a second crossing point appears at lower λ . The two crossing points open up to a V-like structure with increasing disorder.”

In the paper

“We can see how an analog quantum simulator works well when a classical solution is efficient, and worsens (but only in a limited way) when the problem becomes classically hard to simulate.”

In the paper

“For increasing disorder ( r = 0 . 2 , right), the collapse worsens, and the best collapse is obtained for some ν > 1 .”

In the paper

“With increasing disorder ( r = 0 . 2 , right), the crossing point moves to larger values of λ and becomes less well defined. Also, at large disorder there appears a second crossing point below λ = 1 .”

In the paper

“In such a situation, the error correction and fault tolerance are not guaranteed and the question of efficiency is even more pertinent.”

In the paper

“Note specifically that little is known about reliability and efficiency of AQSs (question marks).”

In the paper

“the tools to design, build, and implement QSs [1-3] have rapidly developed and are now reaching very sophisticated levels [4].”

What the sources say

• Not checked yetcites [4] source

In the paper

“Measurements on this engineered machine, now known as a 'quantum simulator,' would reveal some unknown or difficult to compute properties of a model of interest.”

What the sources say

• Not checked yetcites [1] source

In the paper

“The requirements of reliability and efficiency are interrelated.”

In the paper

“modeling interesting physics associated with a quantum system involving 50 spin-1/2 particles - whose general description requires 2 50 ≈ 10 15 complex numbers - is out of the reach of current classical supercomputers, but perhaps within the grasp of a QS.”

In the paper

“In his seminal 1982 paper [1], Richard Feynman suggested that such models might be solved by 'simulation' with a new type of computer whose constituent parts are effectively governed by a desired quantum many-body dynamics.”

What the sources say

• Not checked yetcites [1] source

In the paper

“We argue that a useful quantum simulator must satisfy four conditions: relevance , controllability , reliability , and efficiency .”

In the paper

“They found that for 100 spins, in order to achieve b ≥ 10 bits of precision, at least two levels of concatenated error correction are necessary, requiring at least 100 days of run time on the ion-trap quantum computer; for b ≥ 18 , three levels are necessary, requiring at least 7 . 5 × 10 3 years! These results assume a gate time of 10 µ s .”

What the sources say

• Not checked yetcites [120] source

In the paper

“In fact, we could try to improve the precision of a QS by averaging more experiments, but in hypersensitive regimes (like close to quantum phase transitions) the necessary number of repetitions can grow rapidly, bringing the overall efficiency of the QS down to the level of classical computers.”

In the paper

“In particular, it is possible that for some classes of nUDQSs the problems of Trotterization are not as severe as in the case of UDQSs [101].”

What the sources say

• Not checked yetcites [101] source

In the paper

“In these cases, QMC works for various large systems, but fails for Fermi or frustrated systems due to the famous sign problem [75].”

What the sources say

• Not checked yetcites [75] source

In the paper

“Moreover, Brown et al. showed that for a small number of qubits where one might avoid error correction, analog control errors on the logic gates can lead to faulty results, negating requirement (c), and robust control pulses become essential.”

What the sources say

• Not checked yetcites [64] source

In the paper

“For nUDQSs, it is not guaranteed that error correction and fault-tolerant computing is possible.”

In the paper

“It may be, however, that the universal features shared by multiple platforms are robust only because they could have been predicted efficiently with some classical algorithm.”

In the paper

“The latter statement implies that while it might be possible to simulate with classical computers short-time dynamics in a restricted class of 1D models, such attempts will nearly always fail at longer time scales.”

In the paper

“Perturbation theory works only if there exists a small expansion parameter [76].”

What the sources say

• Not checked yetcites [76] Alba V, 2011, J. Stat. Mech.

In the paper

“Mean field [85], DFT [86, 87], or DMFT [88], finally, have other limitations, e.g., they are essentially designed for weakly-correlated systems.”

What the sources say

• Not checked yetcites [85] Stoudenmire E Wagner L O White S R Burke K, 2011
• Not checked yetcites [86] source
• Not checked yetcites [87] Misguich G, 2004, Frustrated Spin Systems
• Not checked yetcites [88] source

In the paper

“whereas 2D tensor-network methods (PEPS, MERA) in principle work for arbitrarily big systems (bosonic, and even fermionic [83] or frustrated [84]) but are biased towards slightly entangled states.”

What the sources say

• Not checked yetcites [83] Binney J J, 1992, The Theory Of Critical Phenomena : an Introduction to the Renormalization Group source
• Not checked yetcites [84] source

In the paper

“Generally speaking, computationally hard are those 'strongly entangled' models in more than 1D such as”

What the sources say

• Not checked yetcites [68] Lee P A, 2008, Rep. Prog. Phys.
• Not checked yetcites [89] source
• Not checked yetcites [90] source

In the paper

“Indeed, this fact is related to correlation and entanglement spreading according to the Lieb-Robinson theorem that states that, after a sufficiently large time, states can become strongly entangled ([91-96], see also [97]).”

What the sources say

• Not checked yetcites [91] source
• Not checked yetcites [92] source
• Not checked yetcites [93] source
• Not checked yetcites [94] Schuch N, 2008, New J. Phys. source
• Not checked yetcites [95] source
• Not checked yetcites [96] source

In the paper

“Exact diagonalization works only for rather small systems [75].”

What the sources say

• Not checked yetcites [75] source

In the paper

“nUDQSs are typically not perfectly controllable, but in most experimental realizations should allow for a wide control of parameters, which in turn should allow for simulations of evolution for wide families of Hamiltonians of interest.”

In the paper

“In the case of 1D systems, DMRG, MERA and TTN techniques scale favorably and can, in principle, treat very large systems [77-79].”

What the sources say

• Not checked yetcites [77] Evenbly G Vidal G, 2011
• Not checked yetcites [78] source
• Not checked yetcites [79] source

In the paper

“For non-universal AQSs such an addition requires additional resources.”

In the paper

“- Error correction: UAQSs do not allow for standard error correction and fault tolerance.”

In the paper

“The first step toward large-scale QSs with ions was, however, recently achieved [139].”

What the sources say

• Not checked yetcites [139] source

In the paper

“For example, there may be models that have only one known implementation, or different implementations may suffer in the same way from imperfections, hence consistently exhibiting features associated with noise rather than with the ideal model.”

In the paper

“- Error correction: AQSs do not allow for standard error correction and fault tolerance.”

In the paper

“All of these comments imply that in the area of OSDQSs there are more open questions than answers.”

In the paper

“The only difference is that UAQSs, by definition, are capable of performing tests of robustness of the quantum simulations that we propose in the following section, i.e., tests based on adding disorder or noise in a controlled manner to the simulated Hamiltonian.”

In the paper

“While by definition they operate in continuous time and thus the Trotterization problems do not concern them, the standard error-correction methods and fault tolerance cannot be applied.”

In the paper

“While by definition they operate in continuous time and thus the Trotterization problems do not concern them, the standard error-correction methods and fault tolerance cannot be applied.”

In the paper

“Most, if not all of the proposals for and realizations of AQSs allow for at least partial controllability.”

In the paper

“The most advanced experiments with AQSs have been with ultracold atoms in optical lattices [3, 5-15, 19, 71].”

What the sources say

• Not checked yetcites [3] Lewenstein M, 2012, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems source
• Not checked yetcites [5] source
• Not checked yetcites [6] source
• Not checked yetcites [7] source
• Not checked yetcites [8] source
• Not checked yetcites [9] source
• Not checked yetcites [10] source
• Not checked yetcites [11] source
• Not checked yetcites [12] source
• Not checked yetcites [13] source

In the paper

“The degree of quantum control is even better in ultracold-ion systems, but these are so far limited to few ions [21-23, 26, 27].”

What the sources say

• Not checked yetcites [21] source
• Not checked yetcites [22] source
• Not checked yetcites [23] source
• Not checked yetcites [26] source
• Not checked yetcites [27] source

In the paper

“At criticality, the low-energy spectrum of the Hamiltonian is gapless in the thermodynamic limit. In finite systems, it presents non-vanishing gaps that decrease in a systematic way with increasing system size.”

In the paper

“Still, as in the case of OSDQSs, most of these general aspects concerning OSAQSs have not yet been investigated systematically.”

In the paper

“Even more, in this Griffiths phase observables become non-selfaveraging, i.e., fluctuations increase with system size, and hence dominate the thermodynamic limit.”

In the paper

“The TIM, even under the presence of disorder, is efficiently solvable - by which we mean that the eigenstates and eigenenergies of the system can be found using a classical computer, and that the cost of the algorithms (in time and hardware) is polynomial with the size (number of particles) of the system.”

In the paper

“In a universal quantum computation, the fidelity would have to be very close to 1 for the quantum computer to work fault-tolerantly.”

In the paper

“The quantum Ising model, which we describe in detail in Section VI, is exactly solvable, which allows us to explore regions with universal behavior such as second-order QPTs.”

In the paper

“Moreover, a state-of-the-art AQS can achieve very low levels of disorder, whence this is the experimentally relevant regime.”

In the paper

“In practice, we can only deal with finite systems so that we cannot observe real criticality but only smoothed out signatures of it, a phenomenon which one normally calls 'pseudo-criticality.' For example, the correlation length ξ is bounded by the system size. Still, its peak gives a reliable signature for the location of the critical point.”

In the paper

“However, it is well known that even small disorder can lead to novel quantum phases such as Bose [182] or Fermi [183] glasses (see also the section on disorder in Ref. [184] and references therein).”

What the sources say

• Not checked yetcites [182] Cardy J, 2010
• Not checked yetcites [183] source
• Not checked yetcites [184] source

In the paper

“From what we have said, the rigorous answer to this question is 'no', yet in practice we do tend to trust them, at least to some extent.”

In the paper

“Indeed, in the TIM when the disorder strength is comparable with the interactions, the critical point disappears and is replaced by a so called Griffiths phase [180], extending across a region of size proportional to the disorder strength.”

What the sources say

• Not checked yetcites [180] source

In the paper

“The TIM, in particular, can be solved by using a Jordan-Wigner transformation to a system of non-interacting fermions - and the cost of solving the non-interacting fermion system is the cost of diagonalizing a matrix with rank equal to twice the number of spins in the chain [178].”

What the sources say

• Not checked yetcites [178] source

In the paper

“OSAQSs are typically not universal in a sense similar to OSDQSs; they allow neither for simulating arbitrary (local) Markovian dynamics, nor do they allow for preparation of arbitrary states.”

In the paper

“All models within a given universality class give rise to the same collective behavior at large distances (typically large with respect to the lattice spacing), irrespective of their microscopic details [177]. Therefore, all relevant thermodynamic quantities for all models within a class are characterized by the same small set of critical exponents which describe the power-law decay of the”

What the sources say

• Not checked yetcites [177] source

In the paper

“- Realizations: All AQS systems can, in principle, be used as OSAQSs.”

In the paper

“This model is well studied (see for instance [179]): for low fields, the ground state is a ferromagnet, while for large fields it is a paramagnet.”

What the sources say

• Not checked yetcites [179] source

In the paper

“If the QPT is described by a CFT (a specific subclass of one-dimensional critical systems), it is characterized by a central charge c . The central charge appears ubiquitously [187].”

What the sources say

• Not checked yetcites [187] source

In the paper

“We know that for noise above certain levels a digital quantum circuit is classically simulatable and for levels below a certain threshold it can be rendered fault tolerant.”

In the paper

“In the context of many-body physics, the success of DMRG in, e.g., 1D spin chains, is rigorously related to the existence of efficient matrix-product-state representations [197].”

What the sources say

• Not checked yetcites [197] source

In the paper

“For example, models whose collective behavior is that of a free Majorana fermion (as in the disorder-free TIM) have central charge = 1 / 2 , while models whose collective behavior is that of a free boson have central charge = 1 .”

In the paper

“This suggests that such an AQS could perform well in a laboratory demonstration.”

In the paper

“The effect of disorder in both the decay rate and the asymptotic saturation value can, in general, be understood from established techniques such as Fermi Golden's rule, and random matrices [191].”

What the sources say

• Not checked yetcites [191] source

In the paper

“On the other hand, the effect of the initial state and the external driving is known to be nontrivial and of particular interest for our purposes.”

In the paper

“In higher-dimensional lattices, states which obey the socalled 'Area law' [97], where quantum correlations are smaller than in generic states, may still be amenable to a classical simulation using state-of-the-art techniques such as tensor networks [77, 81, 82, 198, 199], Density Functional Theories [86], or Quantum Monte Carlo [75].”

What the sources say

• Not checked yetcites [97] source
• Not checked yetcites [77] Evenbly G Vidal G, 2011
• Not checked yetcites [81] source
• Not checked yetcites [82] Wang L Gu Z Wen X Verstraete F, 2011
• Not checked yetcites [198] source
• Not checked yetcites [199] source
• Not checked yetcites [86] source

In the paper

“This happens, for example, in the realm of digital quantum computation, where a quantum circuit becomes classically simulatable for noise levels above a certain level when quantum gates lose their entangling power [194196].”

What the sources say

• Not checked yetcites [194196]

In the paper

“Strictly speaking, the TIM has an underlying CFT only in the disorder-free case, but there have been efforts to extract an effective central charge also for the disordered model [181], for example, from the von Neumann entropy S of the reduced density matrix of a part of the chain of size l .”

What the sources say

• Not checked yetcites [181] Barber M N, 1983, Phase Transitions and Critical Phenomena

In the paper

“Using the same setup with a global quench, but starting from a thermal initial state, tDMRG becomes efficient for high temperatures [192] where the state and its correlations are almost classical. However, it becomes exponentially inefficient with time for low-temperature initial states.”

What the sources say

• Not checked yetcites [192] source

In the paper

“In fact, in many experimental situations one hopes to reach levels of disorder or noise that are below a few percents, assuring the robustness of the AQS.”

In the paper

“Generically, solving for the dynamics of a quantum manybody system is a hard problem for classical algorithms.”

In the paper

“Hence, ignoring the effects of disorder would give completely erroneous results, since even a small deviation of the central charge indicates a completely different universality class.”

In the paper

“Our main discussion focused on AQSs, but similar issues pertain to DQSs.”

In the paper

“The findings of both rows of figures mean that, ignoring the effects of disorder in an imperfect AQS yields, compared to the ideal model, too large values for the critical exponent as well as the location of the critical point.”

In the paper

“Away from criticality, the correlations decay as C ( i, j ) ∝ exp( -| i -j | /ξ ) with correlation length ξ .”

In the paper

“The correlations are intrinsically connected to the energy gap ∆( L ) , since a gapped system necessarily has exponentially decaying correlations [201].”

What the sources say

• Not checked yetcites [201] Sorella S Becca F 2011

In the paper

“If a given model can be described by a conformal field theory, its universal critical behavior (including all critical exponents) is completely defined by a single number, the central charge c .”

In the paper

“If the initial state is not pure, but a thermal state, the state remains mixed even if the evolution is unitary. In this case, we cannot compute the fidelity for the full many-body state, but only the fidelity of the reduced density matrix for a few spins.”

In the paper

“Without disorder, <!-- formula-not-decoded --> for some universal function f [177]. Hence, one can extract the critical exponent for the correlation length ν from a data collapse of the correlations.”

What the sources say

• Not checked yetcites [177] source

In the paper

“For computing fidelities, we use the convenient Levitov's formula [204, 205], which relates traces of operators in the Hilbert space of the fermions to determinants of much smaller matrices (like H ).”

What the sources say

• Not checked yetcites [204] source
• Not checked yetcites [205] Shi Q Li S Zhao J Zhou H, 2009

In the paper

“There, we plot curves 1 / ( L ζ ∆( L )) for closeby chain lengths L , where ζ is the dynamical critical exponent, which for the disorder-free case equals 1.”

In the paper

“For the case of a single spin, the symmetry of the system ensures that at all times the reduced density matrix can be written as ρ = (1+ &lt; σ i x &gt; σ i x ) / 2 . Since &lt; σ i x &gt; = c † i c i , we only need to compute the evolution of the diagonal terms in the G correlation matrix.”

In the paper

“Even though the underlying model is actually solvable, this may be positive evidence for the existence of an intermediate regime of noise, and the efficiency of AQSs in more complex situations.”

In the paper

“As in any quantum algorithm, initialization, evolution of the state, and measurement must be performed efficiently, i.e., with a polynomial use of physical resources (space and time). Digital quantum simulation is no exception.”