Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.08051

The Space Just Above One Clean Qubit

Dale Jacobs, Saeed Adel Mehraban

Published Oct 10, 2024• 0 citations
arXivHTMLPDF

Abstract

Consider the model of computation where we start with two halves of a $2n$-qubit maximally entangled state. We get to apply a universal quantum computation on one half, measure both halves at the end, and perform classical postprocessing. This model, which we call $\frac12$BQP, was defined in STOC 2017 [ABKM17] to capture the power of permutational computations on special input states. As observed in [ABKM17], this model can be viewed as a natural generalization of the one-clean-qubit model (DQC1) where we learn the content of a high entropy input state only after the computation is completed. An interesting open question is to characterize the power of this model, which seems to sit nontrivially between DQC1 and BQP. In this paper, we show that despite its limitations, this model can carry out many well-known quantum computations that are candidates for exponential speed-up over classical computations (and possibly DQC1). In particular, $\frac12$BQP can simulate Instantaneous Quantum Polynomial Time (IQP) and solve the Deutsch-Jozsa problem, Bernstein-Vazirani problem, Simon's problem, and period finding. As a consequence, $\frac12$BQP also solves Order Finding and Factoring outside of the oracle setting. Furthermore, $\frac12$BQP can solve Forrelation and the corresponding oracle problem given by Raz and Tal [RT22] to separate BQP and PH. We also study limitations of $\frac12$BQP and show that similarly to DQC1, $\frac12$BQP cannot distinguish between unitaries which are close in trace distance, then give an oracle separating $\frac12$BQP and BQP. Due to this limitation, $\frac12$BQP cannot obtain the quadratic speedup for unstructured search given by Grover's algorithm [Gro96]. We conjecture that $\frac12$BQP cannot solve $3$-Forrelation.

DOI
10.48550/arxiv.2410.08051

Related papers

IQP circuits for 2-Forrelation

Quentin Buzet, André Chailloux

The $2$-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating $\mathsf{BQP}$ and $\mathsf{PH}$ relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that $2$-Forrelation can be solved using Instantaneous Quantum Polynomial-time ($\mathsf{IQP}$) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed $2$-Forrelation can be solved by a classical random choice between two one-query $\mathsf{IQP}$ circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle $O$ such that $\mathsf{IQP}^O \not\subseteq \mathsf{PH}^O$, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between $\mathsf{IQP}$ and $\mathsf{DQC}_1$. We prove Fourier growth bounds for multi-query $\mathsf{IQP}$ circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted $\mathsf{IQP}$ model. The key ingredient is an algebraic identity of the quadratic function $Q(x) = \sum_{i < j} x_ix_j$ that allows extracting inner-product phases within an $\mathsf{IQP}$ circuit.

quant-pharXiv:2604.15248•Apr 17, 2026•0 citations

Oracle separations of hybrid quantum-classical circuits

Atul Singh Arora, Alexandru Gheorghiu, Singh, Uttam

An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d $\nsubseteq$ QC_d and QC_d $\nsubseteq$ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.

arXiv:2201.01904•Jan 6, 2022•3 citations

On the Need for Large Quantum Depth

Nai-Hui Chia, Kai-Min Chung, Ching–Yi Lai

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.

arXiv:1909.10303•Journal of the ACM•Nov 23, 2022•11 citations

The complexity of NISQ

Sitan Chen, Jordan S. Cotler, Hsin-Yuan Huang +1 more

The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon's problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.

arXiv:2210.07234•Nature Communications•Sep 26, 2023•101 citations

Oracle Separations in the Fourier Hierarchy

Atul Mantri

The Fourier hierarchy $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\ge2$. We prove that for every constant $k\ge2$ there is an oracle relative to which $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.

quant-pharXiv:2609.11830•Sep 11, 2026•0 citations

Entanglement and coherence in the Bernstein-Vazirani algorithm

Moein Naseri, Tulja Varun Kondra, Suchetana Goswami +2 more

Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.

arXiv:2205.13610•Physical Review A•Dec 21, 2022•35 citations
Consider the model of computation where we start with two halves of a 2n2n2n-qubit maximally entangled state. We get to apply a universal quantum computation on one half, measure both halves at the end, and perform classical postprocessing. This model, which we call 12\frac1221​BQP, was defined in STOC 2017 [ABKM17] to capture the power of permutational computations on special input states. As observed in [ABKM17], this model can be viewed as a natural generalization of the one-clean-qubit model (DQC1) where we learn the content of a high entropy input state only after the computation is completed. An interesting open question is to characterize the power of this model, which seems to sit nontrivially between DQC1 and BQP. In this paper, we show that despite its limitations, this model can carry out many well-known quantum computations that are candidates for exponential speed-up over classical computations (and possibly DQC1). In particular, 12\frac1221​BQP can simulate Instantaneous Quantum Polynomial Time (IQP) and solve the Deutsch-Jozsa problem, Bernstein-Vazirani problem, Simon's problem, and period finding. As a consequence, 12\frac1221​BQP also solves Order Finding and Factoring outside of the oracle setting. Furthermore, 12\frac1221​BQP can solve Forrelation and the corresponding oracle problem given by Raz and Tal [RT22] to separate BQP and PH. We also study limitations of 12\frac1221​BQP and show that similarly to DQC1, 12\frac1221​BQP cannot distinguish between unitaries which are close in trace distance, then give an oracle separating 12\frac1221​BQP and BQP. Due to this limitation, 12\frac1221​BQP cannot obtain the quadratic speedup for unstructured search given by Grover's algorithm [Gro96]. We conjecture that 12\frac1221​BQP cannot solve 333-Forrelation.
The 222-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP\mathsf{BQP}BQP and PH\mathsf{PH}PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 222-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP\mathsf{IQP}IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 222-Forrelation can be solved by a classical random choice between two one-query IQP\mathsf{IQP}IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle OOO such that IQPO⊈PHO\mathsf{IQP}^O \not\subseteq \mathsf{PH}^OIQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP\mathsf{IQP}IQP and DQC1\mathsf{DQC}_1DQC1​. We prove Fourier growth bounds for multi-query IQP\mathsf{IQP}IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP\mathsf{IQP}IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixjQ(x) = \sum_{i < j} x_ix_jQ(x)=∑i<j​xi​xj​ that allows extracting inner-product phases within an IQP\mathsf{IQP}IQP circuit.
An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d ⊈\nsubseteq⊈ QC_d and QC_d ⊈\nsubseteq⊈ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.
The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon's problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.
The Fourier hierarchy FH0⊆FH1⊆FH2⊆⋯\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdotsFH0​⊆FH1​⊆FH2​⊆⋯, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly BPP\mathrm{BPP}BPP, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level k≥2k\ge2k≥2. We prove that for every constant k≥2k\ge2k≥2 there is an oracle relative to which FHk⊊FHk+1\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}FHk​⊊FHk+1​. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level kkk it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from BQP\mathrm{BQP}BQP relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with kkk layers is reproduced exactly by an algorithm making only k−1k-1k−1 rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.
Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.