arXiv subjects

Harry Buhrman

Publications and source records attributed to Harry Buhrman.

At least 19 recordsLinked to original sources

arXiv2025

Unconditional and exponentially large violation of classicality

Testing the predictions of quantum mechanics has been one of the main experimental endeavors for decades. Recent advancements in technology led to a number of demonstrations which test non-classicality via specific computational tasks. Limitations of these experiments include dependence on complexity theory assumptions, susceptibility to hardware noise and inefficient verification, raising questions about their scalability. We propose to test non-classicality using a game based on complement sampling, an efficiently verifiable problem that achieves the largest possible separation between quantum and classical computation when both input and output represent samples from probability distributions. When restricting the input to instances inspired by the Bernstein-Vazirani problem, our game admits an exponentially large violation of classicality without relying on computational hardness assumptions. We execute the game on Quantinuum System Model H2 trapped-ion quantum computers, with experiments consisting of thousands of different circuits on up to 55 qubits. The observed scores can be explained by a systematic adoption of a quantum strategy, further corroborating the quantum nature of the hardware in an efficient and scalable way.

quant-ph↗

arXiv2025

A Simpler Exponential-Time Approximation Algorithm for MAX-k-SAT

We present an extremely simple polynomial-space exponential-time $(1-\varepsilon)$-approximation algorithm for MAX-k-SAT that is (slightly) faster than the previous known polynomial-space $(1-\varepsilon)$-approximation algorithms by Hirsch (Discrete Applied Mathematics, 2003) and Escoffier, Paschos and Tourniaire (Theoretical Computer Science, 2014). Our algorithm repeatedly samples an assignment uniformly at random until finding an assignment that satisfies a large enough fraction of clauses. Surprisingly, we can show the efficiency of this simpler approach by proving that in any instance of MAX-k-SAT (or more generally any instance of MAXCSP), an exponential number of assignments satisfy a fraction of clauses close to the optimal value.

cs.DS↗

arXiv2025

Formal Framework for Quantum Advantage

Motivated by notions of quantum heuristics and by average-case rather than worst-case algorithmic analysis, we define quantum computational advantage in terms of individual problem instances. Inspired by the classical notions of Kolmogorov complexity and instance complexity, we define their quantum versions. This allows us to define queasy instances of computational problems, like e.g. Satisfiability and Factoring, as those whose quantum instance complexity is significantly smaller than their classical instance complexity. These instances indicate quantum advantage: they are easy to solve on a quantum computer, but classical algorithms struggle (they feel queasy). Via a reduction from Factoring, we prove the existence of queasy Satisfiability instances; specifically, these instances are maximally queasy (under reasonable complexity-theoretic assumptions). Further, we show that there is exponential algorithmic utility in the queasiness of a quantum algorithm. This formal framework serves as a beacon that guides the hunt for quantum advantage in practice, and moreover, because its focus lies on single instances, it can lead to new ways of designing quantum algorithms.

quant-ph↗

arXiv2025

Beating the natural Grover bound for low-energy estimation and state preparation

Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this work, we give quantum algorithms which, given any $k$-body Hamiltonian $H$, compute an estimate for the ground state energy and prepare a quantum state achieving said energy, respectively. Specifically, for any $\varepsilon>0$, our algorithms return, with high probability, an estimate of the ground state energy of $H$ within additive error $\varepsilon M$, or a quantum state with the corresponding energy. Here, $M$ is the total strength of all interaction terms, which in general is extensive in the system size. Our approach makes no assumptions about the geometry or spatial locality of interaction terms of the input Hamiltonian and thus handles even long-range or all-to-all interactions, such as in quantum chemistry, where lattice-based techniques break down. In this fully general setting, the runtime of our algorithms scales as $2^{cn/2}$ for $c<1$, yielding the first quantum algorithms for low-energy estimation breaking a standard square root Grover speedup for unstructured search. The core of our approach is remarkably simple, and relies on showing that an extensive fraction of the interactions can be neglected with a controlled error. What this ultimately implies is that even arbitrary $k$-local Hamiltonians have structure in their low energy space, in the form of an exponential-dimensional low energy subspace.

quant-ph↗

arXiv2025

Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the $q$-local Hamiltonian problem with constant promise gap can be solved in $\mathsf{QCMA}$, then $\mathsf{QPCP}[q] \subseteq \mathsf{QCMA}$ for any $q \in O(1)$. (iii) If $\mathsf{QMA}(k)$ has a quantum PCP for any $k \leq \text{poly}(n)$, then $\mathsf{QMA}(2) = \mathsf{QMA}$, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exist (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

quant-ph↗

arXiv2025

Quantum Catalytic Space

Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource. Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of ``dirty'' qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.

quant-ph↗

arXiv2025

Provable and Verifiable Quantum Advantage in Sample Complexity

Consider a fixed universe of $N=2^n$ elements and the uniform distribution over elements of some subset of size $K$. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset. We give a simple quantum algorithm that uses only a single quantum sample -- a single copy of the uniform superposition over elements of the subset. When $K=N/2$, we show that the quantum algorithm succeeds with probability $1$, whereas any classical algorithm that succeeds with bounded probability of error requires a number of samples of the order of $N$. This shows that in a sample-to-sample setting, quantum computation can achieve the largest possible separation over classical computation. We show that the same bound can be lifted to prove average-case hardness, paving the way for demonstrations on noisy intermediate-scale quantum (NISQ) computers. It follows that under the assumption of the existence of one-way functions, complement sampling gives provable, verifiable and NISQable quantum advantage in a sample complexity setting.

quant-ph↗

arXiv2025

Towards experimental demonstration of quantum position verification using true single photons

The geographical position can be a good credential for authentication of a party, this is the basis of position-based cryptography - but classically this cannot be done securely without physical exchange of a private key. However, recently, it has been shown that by combining quantum mechanics with the speed of light limit of special relativity, this might be possible: quantum position verification. Here we demonstrate experimentally a protocol that uses two-photon Hong-Ou-Mandel interference at a beamsplitter, which, in combination with two additional beam splitters and 4 detectors is rendering the protocol resilient to loss. With this we are able to show first results towards an experimental demonstration of quantum position verification.

quant-ph↗

arXiv2024

State preparation by shallow circuits using feed forward

In order to achieve fault-tolerant quantum computation, we need to repeat the following sequence of four steps: First, perform 1 or 2 qubit quantum gates (in parallel if possible). Second, do a syndrome measurement on a subset of the qubits. Third, perform a fast classical computation to establish which errors have occurred (if any). Fourth, depending on the errors, we apply a correction step. Then the procedure repeats with the next sequence of gates. In order for these four steps to succeed, we need the error rate of the gates to be below a certain threshold. Unfortunately, the error rates of current quantum hardware are still too high. On the other hand, current quantum hardware platforms are designed with these four steps in mind. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, constant-depth, quantum circuits that perform 1 qubit gates and nearest-neighbor 2 qubit gates. To explore how this can be useful, we study a computational model which we call Local Alternating Quantum Classical Computations (LAQCC). In this model, qubits are placed in a grid allowing nearest neighbor interactions; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes to control future quantum operations. This model fits naturally between quantum algorithms in the NISQ era and full fledged fault-tolerant quantum computation. We show that LAQCC circuits can create long-ranged interactions, which constant-depth quantum circuits cannot achieve, and use it to construct a range of useful multi-qubit gates. With these gates, we create three new state preparation protocols for a uniform superposition over an arbitrary number of states, W-states, Dicke states and may-body scar states.

quant-ph↗

arXiv2024

Challenges and Opportunities in Quantum Optimization

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

quant-ph↗

arXiv2024

Classical versus quantum queries in quantum PCPs with classical proofs

We generalize quantum-classical PCPs, first introduced by Weggemans, Folkertsma and Cade (TQC 2024), to allow for $q$ quantum queries to a polynomially-sized classical proof ($\mathsf{QCPCP}_{Q,c,s}[q]$). Exploiting a connection with the polynomial method, we prove that for any constant $q$, promise gap $c-s = Ω(1/\text{poly}(n))$ and $δ>0$, we have $\mathsf{QCPCP}_{Q,c,s}[q] \subseteq \mathsf{QCPCP}_{1-δ,1/2+δ}[3] \subseteq \mathsf{BQ} \cdot \mathsf{NP}$, where $\mathsf{BQ} \cdot \mathsf{NP}$ is the class of promise problems with quantum reductions to an $\mathsf{NP}$-complete problem. Surprisingly, this shows that we can amplify the promise gap from inverse polynomial to constant for constant query quantum-classical PCPs, and that any quantum-classical PCP making any constant number of quantum queries can be simulated by one that makes only three classical queries. Nevertheless, even though we can achieve promise gap amplification, our result also gives strong evidence that there exists no constant query quantum-classical PCP for $\mathsf{QCMA}$, as it is unlikely that $\mathsf{QCMA} \subseteq \mathsf{BQ} \cdot \mathsf{NP}$, which we support by giving oracular evidence. In the (poly-)logarithmic query regime, we show for any positive integer $c$, there exists an oracle relative to which $\mathsf{QCPCP}[\mathcal{O}(\log^c n)] \subsetneq \mathsf{QCPCP}_Q[\mathcal{O}(\log^c n)]$, contrasting the constant query case where the equivalence of both query models holds relative to any oracle. Finally, we connect our results to more general quantum-classical interactive proof systems.

quant-ph↗

arXiv2024

Relating non-local quantum computation to information theoretic cryptography

Non-local quantum computation (NLQC) is a cheating strategy for position-verification schemes, and has appeared in the context of the AdS/CFT correspondence. Here, we connect NLQC to the wider context of information theoretic cryptography by relating it to a number of other cryptographic primitives. We show one special case of NLQC, known as $f$-routing, is equivalent to the quantum analogue of the conditional disclosure of secrets (CDS) primitive, where by equivalent we mean that a protocol for one task gives a protocol for the other with only small overhead in resource costs. We further consider another special case of position verification, which we call coherent function evaluation (CFE), and show CFE protocols induce similarly efficient protocols for the private simultaneous message passing (PSM) scenario. By relating position-verification to these cryptographic primitives, a number of results in the cryptography literature give new implications for NLQC, and vice versa. These include the first sub-exponential upper bounds on the worst case cost of $f$-routing of $2^{O(\sqrt{n\log n})}$ entanglement, the first example of an efficient $f$-routing strategy for a problem believed to be outside $P/poly$, linear lower bounds on entanglement for CDS in the quantum setting, linear lower bounds on communication cost of CFE, and efficient protocols for CDS in the quantum setting for functions that can be computed with quantum circuits of low $T$ depth.

quant-ph↗

arXiv2024

Permutation tests for quantum state identity

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether $n$ unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or identical. Under the one-sided error requirement, it is known that the permutation test is optimal for this task, and for two input states this coincides with the well-known Swap test. Until now, the optimal measurement in the general two-sided error regime was unknown. Under more specific promises, the problem can be solved approximately or even optimally with simpler tests, such as the circle test. This work attempts to capture the underlying structure of the quantum state identity problem. Using tools from semidefinite programming and representation theory, we (i) give an optimal test for any input distribution without the one-sided error requirement by writing the problem as an SDP, giving the exact solutions to the primal and dual programs and showing that the two values coincide; (ii) propose a general $G$-test which uses an arbitrary subgroup $G$ of $\text{S}_n$, giving an analytic expression of the performance of the specific test, and (iii) give an approximation of the permutation test using only a classical permutation and $n-1$ Swap tests.

quant-ph↗

arXiv2023

Making Existing Quantum Position Verification Protocols Secure Against Arbitrary Transmission Loss

Signal loss poses a significant threat to the security of quantum cryptography when the chosen protocol lacks loss-tolerance. In quantum position verification (QPV) protocols, even relatively small loss rates can compromise security. The goal is thus to find protocols that remain secure under practically achievable loss rates. In this work, we modify the usual structure of QPV protocols and prove that this modification makes the potentially high transmission loss between the verifiers and the prover security-irrelevant for a class of protocols that includes a practically-interesting candidate protocol inspired by the BB84 protocol ($\mathrm{QPV}_{\mathrm{BB84}}^{f}$). This modification, which involves photon presence detection, a small time delay at the prover, and a commitment to play before proceeding, reduces the overall loss rate to just the prover's laboratory. The adapted protocol c-$\mathrm{QPV}_{\mathrm{BB84}}^{f}$ then becomes a practically feasible QPV protocol with strong security guarantees, even against attackers using adaptive strategies. As the loss rate between the verifiers and prover is mainly dictated by the distance between them, secure QPV over longer distances becomes possible. We also show possible implementations of the required photon presence detection, making c-$\mathrm{QPV}_{\mathrm{BB84}}^{f}$ a protocol that solves all major practical issues in QPV. Finally, we discuss experimental aspects and give parameter estimations.

quant-ph↗

arXiv2023

Noisy decoding by shallow circuits with parities: classical and quantum

We consider the problem of decoding corrupted error correcting codes with NC$^0[\oplus]$ circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability $Ω(\varepsilon^2)$ even if a $(1/2 - \varepsilon)$-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate ``poor man's cat states'' by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.

cs.CC↗

arXiv2023

A Qubit, a Coin, and an Advice String Walk Into a Relational Problem

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly != FBQP/poly, unconditionally, with no oracle -- a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP is not contained in FP/poly -- that is, Adleman's Theorem fails for relational problems -- unless PSPACE is contained in NP/poly. Our proof uses IP=PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP not in FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: * Unconditionally, FP != FBPP and FP/poly != FBPP/poly (even when these classes are carefully defined). * FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly != SampBPP/rpoly (and likewise for SampBQP).

quant-ph↗

arXiv2022

Quantum majority vote

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state $|ψ_1\rangle \otimes \dots \otimes |ψ_n\rangle$ where each qubit is in one of two orthogonal states $|ψ\rangle$ or $|ψ^\perp\rangle$, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of $1/2 + Θ(1/\sqrt{n})$. Under the promise that at least $2/3$ of the input qubits are in the majority state, the fidelity increases to $1 - Θ(1/n)$ and approaches $1$ as $n$ increases. We also consider the more general problem of computing any symmetric and equivariant Boolean function $f: \{0,1\}^n \to \{0,1\}$ in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size $O(n)$. The time complexity of the algorithm is $O(n^4 \log n)$ where $n$ is the number of input qubits.

quant-ph↗

arXiv2022

Towards practical and error-robust quantum position verification

Loss of inputs can be detrimental to the security of quantum position verification (QPV) protocols, as it may allow attackers to not answer on all played rounds, but only on those they perform well on. In this work, we study \textit{loss-tolerant} QPV protocols. We propose a new fully loss-tolerant protocol QPV$_{\textsf{SWAP}}$, based on the SWAP test, with several desirable properties. The task of the protocol, which could be implemented using only a single beam splitter and two detectors, is to estimate the overlap between two input states. By formulating possible attacks as a semi-definite program (SDP), we prove full loss tolerance against unentangled attackers restricted to local operations and classical communication, and show that the attack probability decays exponentially under parallel repetition of rounds. We show that the protocol remains secure even if unentangled attackers are allowed to quantum communicate, making our protocol the first fully loss-tolerant protocol with this property. A detailed analysis under experimental conditions is conducted, showing that QPV$_{\textsf{SWAP}}$ remains fairly robust against equipment errors. We identify a necessary condition for security with errors and simulate one instance of our protocol with currently realistic experimental parameters, gathering that an attack success probability of $\leq10^{-6}$ can be achieved by collecting just a few hundred conclusive protocol rounds.

quant-ph↗