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Dorit Aharonov

Publications and source records attributed to Dorit Aharonov.

At least 37 records · Page 2Linked to original sources
arXiv2018

Quantum Circuit Depth Lower Bounds For Homological Codes

We provide an $Ω(log(n))$ lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code. No circuit-depth lower bound was known before on this code in the general case where the gates can connect qubits even if they are far away; To the best of our knowledge, this is the first time a quantum circuit-depth lower bound is given for unique ground state of a {\it gapped} local Hamiltonian. Providing a lower bound in this case seems more challenging, since such systems exhibit exponential decay of correlations and standard lower bound techniques do not apply. We prove our lower bound by introducing the new notion of $γ$-separation, and analyzing its behavior using algebraic topology arguments. We extend out methods also to a wide class of polygonal complexes beyond the sphere, and prove a circuit-depth lower bound whenever the complex does not have a small "bottle neck" (in a sense which we define). Here our lower bound on the circuit depth is only $Ω(logloglog(n))$. We conjecture that the correct lower bound is at least $Ω(log(log(n))$, but this seems harder to achieve due to the possibility of hyperbolic geometry. For general simplicial complexes the lack of geometrical restriction on the gates becomes considerably more problematic than for the sphere, and we need to thoroughly modify the original argument in order to get a meaningful bound. To the best of our knowledge, this is the first time the class of trivial quantum states is separated from the class of unique ground states of gapped local Hamiltonians; we provide a survey of the current status of this hierarchy for completeness.The tools developed here will be useful in various contexts in which quantum circuit depth lower bounds are of interest, including the study of topological order, quantum computational complexity and quantum algorithmic speedups.

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arXiv2018

Hamiltonian sparsification and gap-simulations

Analog quantum simulations---simulations of one Hamiltonian by another---is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian $\tilde{H}$ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian $H$ (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where $\tilde{H}$ is only required to simulate the groundstate(s) and spectral gap of $H$ instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves a good starting point for future research of these topics.

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arXiv2018

On the Complexity of Two Dimensional Commuting Local Hamiltonians

The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are $k$-local CLHs, for any constant $k$; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of $4$-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.

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arXiv2018

A Quantum inspired proof of $P^{\#P} \subseteq IP$

We provide a new, quantum inspired, proof for the celebrated claim of [LFKN92] that $P^{\#P} \subseteq IP$. The protocol is fundamentally different from the original sum-check protocol of [LFKN92, Sha92], as well as from variants of this proof [GKR08, Mei09], though it still possesses the overall structure of inductively checking consistency between subsequent step. The protocol is inspired by [AAV13]. Hopefully, this protocol will be helpful in making progress towards two major open problems: resolving the quantum PCP question, and verification of quantum computations using a classical BPP verifier. Given the historical importance of the sum-check protocol in classical computational complexity, we hope that the new protocol will also find applications in classical complexity theory.

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arXiv2017

Dining Philosophers, Leader Election and Ring Size problems, in the quantum setting

We provide the first quantum (exact) protocol for the Dining Philosophers problem (DP), a central problem in distributed algorithms. It is well known that the problem cannot be solved exactly in the classical setting. We then use our DP protocol to provide a new quantum protocol for the tightly related problem of exact leader election (LE) on a ring, improving significantly in both time and memory complexity over the known LE protocol by Tani et. al. To do this, we show that in some sense the exact DP and exact LE problems are equivalent; interestingly, in the classical non-exact setting they are not. Hopefully, the results will lead to exact quantum protocols for other important distributed algorithmic questions; in particular, we discuss interesting connections to the ring size problem, as well as to a physically motivated question of breaking symmetry in 1D translationally invariant systems.

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arXiv2017

Interactive Proofs for Quantum Computations

The widely held belief that BQP strictly contains BPP raises fundamental questions: if we cannot efficiently compute predictions for the behavior of quantum systems, how can we test their behavior? In other words, is quantum mechanics falsifiable? In cryptographic settings, how can a customer of a future untrusted quantum computing company be convinced of the correctness of its quantum computations? To provide answers to these questions, we define Quantum Prover Interactive Proofs (QPIP). Whereas in standard interactive proofs the prover is computationally unbounded, here our prover is in BQP, representing a quantum computer. The verifier models our current computational capabilities: it is a BPP machine, with access to only a few qubits. Our main theorem states, roughly: 'Any language in BQP has a QPIP, which also hides the computation from the prover'. We provide two proofs, one based on a quantum authentication scheme (QAS) relying on random Clifford rotations and the other based on a QAS which uses polynomial codes (BOCG+ 06), combined with secure multiparty computation methods. This is the journal version of work reported in 2008 (ABOE08) and presented in ICS 2010; here we have completed the details and made the proofs rigorous. Some of the proofs required major modifications and corrections. Notably, the claim that the polynomial QPIP is fault tolerant was removed. Similar results (with different protocols) were reported independently around the same time of the original version in BFK08. The initial independent works (ABOE08, BFK08) ignited a long line of research of blind verifiable quantum computation, which we survey here, along with connections to various cryptographic problems. Importantly, the problems of making the results fault tolerant as well as removing the need for quantum communication altogether remain open.

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arXiv2017

Fast-forwarding of Hamiltonians and Exponentially Precise Measurements

In the early days of quantum mechanics, it was believed that the time energy uncertainty principle (TEUP) bounds the efficiency of energy measurements, relating the duration ($Δt$) of the measurement, and its accuracy error ($ΔE$) by $ΔtΔE \ge$ 1/2. In 1961 Y. Aharonov and Bohm gave a counterexample, whereas Aharonov, Massar and Popescu [2002] showed that under certain conditions the principle holds. Can we classify when and to what extent the TEUP is violated? Our main theorem asserts that such violations are in one to one correspondence with the ability to "fast forward" the associated Hamiltonian, namely, to simulate its evolution for time $t$ using much less than $t$ quantum gates. This intriguingly links precision measurements with quantum algorithms. Our theorem is stated in terms of a modified TEUP, which we call the computational TEUP (cTEUP). In this principle the time duration ($Δt$) is replaced by the number of quantum gates required to perform the measurement, and we argue why this is more suitable to study if one is to understand the totality of physical resources required to perform an accurate measurement. The inspiration for this result is a family of Hamiltonians we construct, based on Shor's algorithm, which exponentially violate the cTEUP (and the TEUP), thus allowing exponential fast forwarding. We further show that commuting local Hamiltonians and quadratic Hamiltonians of fermions (e.g., Anderson localization model), can be fast forwarded. The work raises the question of finding a physical criterion for fast forwarding, in particular, can many body localization systems be fast forwarded? We rule out a general fast-forwarding method for all physically realizable Hamiltonians (unless BQP=PSPACE). Connections to quantum metrology and to Susskind's complexification of a wormhole's length are discussed.

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arXiv2015

Quantum Error-Correction-Enhanced Magnetometer Overcoming the Limit Imposed by Relaxation

When incorporated in quantum sensing protocols, quantum error correction can be used to correct for high frequency noise, as the correction procedure does not depend on the actual shape of the noise spectrum. As such, it provides a powerful way to complement usual refocusing techniques. Relaxation imposes a fundamental limit on the sensitivity of state of the art quantum sensors which cannot be overcome by dynamical decoupling. The only way to overcome this is to utilize quantum error correcting codes. We present a superconducting magnetometry design that incorporates approximate quantum error correction, in which the signal is generated by a two qubit Hamiltonian term. This two-qubit term is provided by the dynamics of a tunable coupler between two transmon qubits. For fast enough correction, it is possible to lengthen the coherence time of the device beyond the relaxation limit.

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arXiv2014

Local tests of global entanglement and a counterexample to the generalized area law

We introduce a technique for applying quantum expanders in a distributed fashion, and use it to solve two basic questions: testing whether a bipartite quantum state shared by two parties is the maximally entangled state and disproving a generalized area law. In the process these two questions which appear completely unrelated turn out to be two sides of the same coin. Strikingly in both cases a constant amount of resources are used to verify a global property.

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arXiv2014

A simpler proof of existence of quantum weak coin flipping with arbitrarily small bias

Mochon's proof [Moc07] of existence of quantum weak coin flipping with arbitrarily small bias is a fundamental result in quantum cryptography, but at the same time one of the least understood. Though used several times as a black box in important follow-up results [Gan09, CK09, AS10, CK11, KZ13] the result has not been peer-reviewed, its novel techniques (and in particular Kitaev's point game formalism) have not been applied anywhere else, and an explicit protocol is missing. We believe that truly understanding the existence proof and the novel techniques it relies on would constitute a major step in quantum information theory, leading to deeper understanding of entanglement and of quantum protocols in general. In this work, we make a first step in this direction. We simplify parts of Mochon's construction considerably, making about 20 pages of analysis in the original proof superfluous, clarifying some other parts of the proof on the way, and presenting the proof in a way which is conceptually easier to grasp. We believe the resulting proof of existence is easier to understand, more readable, and certainly verifiable. Moreover, we analyze the resources needed to achieve a bias $ε$ and show that the number of qubits is $O(\log 1/ε)$, while the number of rounds is $(1/ε)^{O(1/ε)}$. A true understanding of the proof, including Kitaev's point game techniques and their applicability, as well as completing the task of constructing an explicit (and also simpler and more efficient) protocol, are left to future work.

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arXiv2013

The commuting local Hamiltonian on locally-expanding graphs is in NP

The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP. The complexity of its semi-classical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locally-expanding graph, namely, the expansion of any constant-size set is e-close to optimal, then approximating its ground energy to within additive factor O(e) lies in NP. The proof holds for k- local Hamiltonians for any constant k and any constant dimensionality of particles d. We also show that the approximation problem of CLH on such good local expanders is NP-hard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular Dinur's PCP proof) in an attempt to prove the quantum PCP conjecture. A related result was discovered recently independently by Brandao and Harrow, for 2-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NP-hardness is known in that case.

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arXiv2013

Quantum Locally Testable Codes

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, together with some basic results using those definitions. The most crucial parameter of such codes is their soundness, $R(δ)$, namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code ($δ$, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This phenomenon, which can be attributed to monogamy of entanglement, stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on poor small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs.

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arXiv2013

Increasing sensing resolution with error correction

The signal to noise ratio of quantum sensing protocols scales with the square root of the coherence time. Thus, increasing this time is a key goal in the field. Dynamical decoupling has proven to be efficient in prolonging the coherence times for the benefit of quantum sensing. However, dynamical decoupling can only push the sensitivity up to a certain limit. In this work we present a new approach to increasing the coherence time further through error correction which can improve the efficiency of quantum sensing beyond the fundamental limits of current state of the art methods.

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arXiv2013

The Quantum PCP Conjecture

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.

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arXiv2011

Quantum Hamiltonian complexity and the detectability lemma

Quantum Hamiltonian complexity studies computational complexity aspects of local Hamiltonians and ground states; these questions can be viewed as generalizations of classical computational complexity problems related to local constraint satisfaction (such as SAT), with the additional ingredient of multi-particle entanglement. This additional ingredient of course makes generalizations of celebrated theorems such as the PCP theorem from classical to the quantum domain highly non-trivial; it also raises entirely new questions such as bounds on entanglement and correlations in ground states, and in particular area laws. We propose a simple combinatorial tool that helps to handle such questions: it is a simplified, yet more general version of the detectability lemma introduced by us in the more restricted context on quantum gap amplification a year ago. Here, we argue that this lemma is applicable in much more general contexts. We use it to provide a simplified and more combinatorial proof of Hastings' 1D area law, together with a less than 1 page proof of the decay of correlations in gapped local Hamiltonian systems in any constant dimension. We explain how the detectability lemma can replace the Lieb-Robinson bound in various other contexts, and argue that it constitutes a basic tool for the study of local Hamiltonians and their ground states in relation to various questions in quantum Hamiltonian complexity.

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arXiv2011

The BQP-hardness of approximating the Jones Polynomial

A celebrated important result due to Freedman, Larsen and Wang states that providing additive approximations of the Jones polynomial at the k'th root of unity, for constant k=5 and k>6, is BQP-hard. Together with the algorithmic results of Freedman et al and Aharonov et al, this gives perhaps the most natural BQP-complete problem known today and motivates further study of the topic. In this paper we focus on the universality proof; we extend the universality result of Freedman et al to k's that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values for which the AJL algorithm applies, proving that for all those values, the problems are BQP-complete. As a side benefit, we derive a fairly elementary proof of the Freedman et al density result, without referring to advanced results from Lie algebra representation theory, making this important result accessible to computer science audience. We make use of two general lemmas we prove, the Bridge lemma and the Decoupling lemma, which provide tools for establishing density of subgroups in SU(n). Those tools seem to be of independent interest in more general contexts of proving quantum universality. Our result also implies a completely classical statement, that the_multiplicative_ approximations of the Jones polynomial, at exactly the same values, are #P-hard, via a recent result due to Kuperberg. Since the first publication of those results in their preliminary form (arXiv:quant-ph/0605181v2), the methods we present here were used in several other contexts. This paper is an improved and extended version of the original results, and also includes discussions of the developments since then.

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