arXiv subjects

Michael R. Geller

Publications and source records attributed to Michael R. Geller.

At least 19 recordsLinked to original sources

arXiv2026

Emergent Non-Markovian Nonlinear Qubit From Collective Spin Interactions

Open-system descriptions are typically introduced by coupling a quantum system to an external environment. Here we show that a closed interacting many-body system can itself generate a controlled non-Markovian quantum channel acting on a reduced nonlinear qubit through finite-size corrections to a nonlinear mean-field limit. We demonstrate this using the Kitagawa-Ueda one-axis twisting model, $H=χJ_z^2$, a paradigmatic model of collective spin dynamics, spin squeezing, and two-component Bose-Einstein condensates. Although the large-$N$ regime of this model has been extensively studied, the conventional fixed-$χ$ scaling does not yield a nontrivial dynamical large-$N$ limit. In this paper, we investigate a complementary large-$N$ formulation obtained from the double limit $N\rightarrow\infty$ and $χ\rightarrow O(g/N)$, where $g$ is a coupling constant. We derive the leading finite-$N$ corrections to this limit and show that they correspond to an emergent non-Markovian dephasing process, producing a Gaussian decay of the Bloch-vector coherence with characteristic timescale $t_φ\geq\sqrt{N}/(2g)$. Exact finite-$N$ calculations demonstrate that this effective open-system description becomes quantitatively accurate for systems containing on the order of one hundred qubits. The resulting framework provides a microscopic realization of non-Markovian dephasing generated intrinsically by a closed many-body system and enables efficient simulation of collective quantum dynamics beyond unitary mean-field theory. These results link the long-studied phenomenon of phase diffusion in atomic ensembles and Bose-Einstein condensates to the growing effort to characterize non-Markovian, beyond-Lindblad noise in quantum computing hardware, providing a rare case in which such a noise channel is derived from microscopic dynamics rather than fit phenomenologically.

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arXiv2026

Quantum algorithm for Valiant-Vazirani reduction

There is growing interest in extensions of the standard model of gate-based quantum computation to include auxiliary degrees of freedom evolving according to a nonlinear Schrödinger equation. By reducing the Boolean satisfiability problem SAT to quantum state discrimination, Abrams and Lloyd argued that the right type of nonlinearity can be used to solve NP and #P problems in polynomial time, at least in an idealized noise-free limit. For practical implementation, however, we are restricted to simulated and emergent nonlinearities, such as that appearing in mean field models for ultracold atoms and similar ensembles. A prominent example is the torsion model, which arises in two-component Bose-Einstein condensates and spin models with all-to-all Ising interaction. But torsion-based state discrimination appears to fall short of solving SAT. Here we close this gap by constructing the filtered oracle of the Valiant-Vazirani theorem, providing a randomized polynomial-time reduction from SAT to UNIQUE SAT, a promise problem where there is at most 1 satisfying assignment. In the noise-free limit, the UNIQUE SAT problem can be solved in polynomial time using torsion nonlinearity. Quantum Valiant-Vazirani reduction is no faster than the efficient classical version, but a fault-tolerant implementation coupled to a nonlinear quantum coprocessor simulating torsion would enable polynomial time solution to NP (but not #P) problems.

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arXiv2026

Nonlinear Hamiltonians and Boolean satisfiability

We consider an extended model of quantum computation where a scalable fault-tolerant quantum computer is coupled to one or more ancilla qubits that evolve according to a nonlinear Schrödinger equation. Following the approach of Abrams and Lloyd, an efficient quantum circuit evaluating an $n$-bit Boolean function in conjunctive normal form is used to prepare an ancilla encoding its number $s$ of satisfying assignments ($0 \le s \le 2^n$). This is followed by a nonlinear quantum state discrimination gate on the ancilla qubit that is used to learn properties of $s$. Here we consider three types of state discriminators generated by different nonlinear Hamiltonians. First, given a restricted Boolean satisfiability problem with the promise of at most one satisfying assignment ($ 0 \le s \le 1$), we show that a qubit with $\langle σ^z \rangle σ^z$ nonlinearity can be used to efficiently determine whether $s = 0$ or $s = 1$, solving the UNIQUE SAT problem. Here $\langle A \rangle := \langle ψ| A |ψ\rangle $ denotes expectation in the current state. UNIQUE SAT is NP-hard under a randomized polynomial-time reduction (of course any discussion of complexity assumes a scalable, fault-tolerant implementation). Second, for unrestricted satisfiability problems with $ 0 \le s \le 2^n$, a Hamiltonian with $ \langle σ^x \rangle σ^y - \langle σ^y \rangle σ^x$ nonlinearity can be used to efficiently determine whether $s=0$ or $s>0$, thereby solving 3SAT, which is NP-complete. Finally, we show that $ \langle σ^y \rangle \langle σ^z \rangle σ^x - \langle σ^x \rangle \langle σ^z \rangle σ^y $ nonlinearity can be used to efficiently measure $s$ and solve #SAT, which is #P-complete. The nonlinear models are of mean field type and might be simulated with ultracold atoms.

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arXiv2026

From spin squeezing to fast state discrimination

There is great interest in generating and controlling entanglement in Bose-Einstein condensates and similar ensembles for use in quantum computation, simulation, and sensing. One class of entangled states useful for enhanced metrology are spin-squeezed states of $N$ two-level atoms. After preparing a spin coherent state of width $1/\sqrt{N}$ centered at coordinates $( θ, ϕ) $ on the Bloch sphere, atomic interactions generate a nonlinear evolution that shears the state's probability density, stretching it to an ellipse and causing squeezing in a direction perpendicular to the major axis. Here we consider the same setup but in the $N \rightarrow \infty $ limit . This shrinks the initial coherent state to zero area. Large $N$ also suppresses two-particle entanglement and squeezing, as required by a monogamy bound. The torsion (1-axis twist) is still present, however, and the center of the large $N$ coherent state evolves as a qubit governed by a two-state Gross-Pitaevskii equation. The resulting nonlinearity is known to be a powerful resource in quantum computation. It can be used to implement single-input quantum state discrimination, an impossibility within linear one-particle quantum mechanics. We obtain a solution to the discrimination problem in terms of a Viviani curve on the Bloch sphere. We also consider an open-system variant containing both Bloch sphere torsion and dissipation. In this case it should be possible to generate two basins of attraction within the Bloch ball, having a shared boundary that can be used for a type of autonomous state discrimination. We explore these and other connections between spin squeezing in the large $N$ limit and nonlinear quantum gates, and argue that a two-component condensate is a promising platform for realizing a nonlinear qubit.

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arXiv2025

Quantum simulation of a qubit with non-Hermitian Hamiltonian

Modeling non-Hermitian Hamiltonians is increasingly important in classical and quantum domains, especially when studying open systems, $PT$ symmetry, and resonances. However, the quantum simulation of these models has been limited by the extensive resources necessary in iterative methods with exponentially small postselection success probability. Here we employ a fixed-depth variational circuit to circumvent these limitations, enabling simulation deep into the $PT$-broken regime surrounding an exceptional point. Quantum simulations are carried out using IBM superconducting qubits. The results underscore the potential for variational quantum circuits and machine learning to push the boundaries of quantum simulation, offering new methods for exploring quantum phenomena with near-term intermediate-scale quantum technology.

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arXiv2024

Protocol for nonlinear state discrimination in rotating condensate

Nonlinear mean field dynamics enables quantum information processing operations that are impossible in linear one-particle quantum mechanics. In this approach, a register of bosonic qubits (such as neutral atoms or polaritons) is initialized into a symmetric product state through condensation, then subsequently controlled by varying the qubit-qubit interaction. We propose an experimental implementation of quantum state discrimination, an important subroutine in quantum computation, with a toroidal Bose-Einstein condensate. The condensed bosons here are atoms, each in the same superposition of angular momenta 0 and 1, encoding a qubit. A nice feature of the protocol is that only readout of individual quantized circulation states (not superpositions) is required.

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arXiv2023

The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model

We investigate models of nonlinear qubit evolution based on mappings to an $n$-qubit central spin model (CSM) in the large $n$ limit, where mean field theory is exact. Extending a theorem of Erdös and Schlein, we establish that the CSM is rigorously dual to a nonlinear qubit when $n \rightarrow \infty$. The duality supports a type of nonlinear quantum computation in systems, such as a condensate, where a large number of ancilla couple symmetrically to a "central" qubit. It also enables a gate-model implementation of nonlinear quantum simulation with a rigorous error bound. Two variants of the model, with and without coupling between ancilla, map to effective models with different nonlinearity and symmetry. The duality discussed here might also be interesting from a quantum foundations perspective. There has long been interest in whether quantum mechanics might possess some type of small, unobserved nonlinearity. If not, what is the principle prohibiting it? The duality implies that there is not a sharp distinction between universes evolving according to linear and nonlinear quantum mechanics: A one-qubit "universe" prepared in a pure state $| φ\rangle $ at the time of the big bang and symmetrically coupled to ancilla prepared in the same state, would appear to evolve nonlinearly for any finite time $t>0$ as long as there are exponentially many ancilla $n \gg {\rm exp}(O(t))$.

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arXiv2023

Proposal for a Lorenz qubit

Nonlinear qubit master equations have recently been shown to exhibit rich dynamical phenomena such as period doubling, Hopf bifurcation, and strange attractors usually associated with classical nonlinear systems. Here we investigate nonlinear qubit models that support tunable Lorenz attractors. A Lorenz qubit could be realized experimentally by combining qubit torsion, generated by real or simulated mean field dynamics, with linear amplification and dissipation. This would extend engineered Lorenz systems to the quantum regime, allowing for their direct experimental study and possible application to quantum information processing.

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arXiv2023

Nonlinear and non-CP gates for Bloch vector amplification

Any state r = (x,y,z) of a qubit, written in the Pauli basis and initialized in the pure state r = (0,0,1), can be prepared by composing three quantum operations: two unitary rotation gates to reach a pure state on the Bloch sphere, followed by a depolarization gate to decrease |r|. Here we discuss the complementary state-preparation protocol for qubits initialized at the center of the Bloch ball, r=0, based on increasing or amplifying |r| to its desired value, then rotating. Bloch vector amplification may or may not increase qubit energy, but it necessarily increases purity and decreases entropy. Amplification can be achieved with a linear Markovian CPTP channel by placing the channel's fixed point away from r=0, making it nonunital, but the resulting gate suffers from a critical slowing down as that fixed point is approached. Here we consider alternative designs based on linear and nonlinear Markovian PTP channels, which offer benefits relative to linear CPTP channels, namely fast Bloch vector amplification without deceleration. These operations simulate a reversal of the thermodynamic arrow of time for the qubit and would provide striking experimental demonstrations of non-CP dynamics.

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arXiv2023

Fast quantum state discrimination with nonlinear PTP channels

We investigate models of nonlinear quantum computation based on deterministic positive trace-preserving (PTP) channels and evolution equations. The models are defined in any finite Hilbert space, but the main results are for dimension $N \! = \! 2$. For every normalizable linear or nonlinear positive map $ϕ$ on bounded linear operators $X$, there is an associated normalized PTP channel $ ϕ(X) / {\rm tr}[ϕ(X)]$. Normalized PTP channels include unitary mean field theories, such as the Gross-Pitaevskii equation for interacting bosons, as well as models of linear and nonlinear dissipation. They classify into 4 types, yielding 3 distinct forms of nonlinearity whose computational power we explore. In the qubit case these channels support Bloch ball torsion and other distortions studied previously, where it has been shown that such nonlinearity can be used to increase the separation between a pair of close qubit states, suggesting an exponential speedup for state discrimination. Building on this idea, we argue that this operation can be made robust to noise by using dissipation to induce a bifurcation to a novel phase where a pair of attracting fixed points create an intrinsically fault-tolerant nonlinear state discriminator.

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arXiv2022

Quantum algorithm for Wang-Landau sampling

It has been shown that the Metropolis algorithm can be implemented on quantum computers in a way that avoids the sign problem. However, flat histogram techniques are often preferred as they don't suffer from the same limitations that afflict Metropolis for problems of real-world interest and provide a host of other benefits. In particular, the Wang-Landau method is known for its efficiency and accuracy. In this work we design, implement, and validate a quantum algorithm for Wang-Landau sampling, greatly expanding the range of quantum many body problems solvable by Monte Carlo simulation.

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arXiv2021

Conditionally rigorous mitigation of multiqubit measurement errors

Several techniques have been recently introduced to mitigate errors in near-term quantum computers without the overhead required by quantum error correcting codes. While most of the focus has been on gate errors, measurement errors are significantly larger than gate errors on some platforms. A widely used {\it transition matrix error mitigation} (TMEM) technique uses measured transition probabilities between initial and final classical states to correct subsequently measured data. However from a rigorous perspective, the noisy measurement should be calibrated with perfectly prepared initial states and the presence of any state-preparation error corrupts the resulting mitigation. Here we develop a measurement error mitigation technique, conditionally rigorous TMEM, that is not sensitive to state-preparation errors and thus avoids this limitation. We demonstrate the importance of the technique for high-precision measurement and for quantum foundations experiments by measuring Mermin polynomials on IBM Q superconducting qubits. An extension of the technique allows one to correct for both state-preparation and measurement (SPAM) errors in expectation values as well; we illustrate this by giving a protocol for fully SPAM-corrected quantum process tomography.

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arXiv2021

Quantum simulation of operator spreading in the chaotic Ising model

There is great interest in using near-term quantum computers to simulate and study foundational problems in quantum mechanics and quantum information science, such as the scrambling measured by an out-of-time-ordered correlator (OTOC). Here we use an IBM Q processor, quantum error mitigation, and weaved Trotter simulation to study high-resolution operator spreading in a 4-spin Ising model as a function of space, time, and integrability. Reaching 4 spins while retaining high circuit fidelity is made possible by the use of a physically motivated fixed-node variant of the OTOC, allowing scrambling to be estimated without overhead. We find clear signatures of ballistic operator spreading in a chaotic regime, as well as operator localization in an integrable regime. The techniques developed and demonstrated here open up the possibility of using cloud-based quantum computers to study and visualize scrambling phenomena, as well as quantum information dynamics more generally.

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arXiv2021

Categorizing Readout Error Correlations on Near Term Quantum Computers

Readout errors are a significant source of noise for near term quantum computers. A variety of methods have been proposed to mitigate these errors using classical post processing. For a system with $n$ qubits, the entire readout error profile is specified by a $2^n\times 2^n$ matrix. Recent proposals to use sub-exponential approximations rely on small and/or short-ranged error correlations. In this paper, we introduce and demonstrate a methodology to categorize and quantify multiqubit readout error correlations. Two distinct types of error correlations are considered: sensitivity of the measurement of a given qubit to the state of nearby "spectator" qubits, and measurement operator covariances. We deploy this methodology on IBMQ quantum computers, finding that error correlations are indeed small compared to the single-qubit readout errors on IBMQ Melbourne (15 qubits) and IBMQ Manhattan (65 qubits), but that correlations on IBMQ Melbourne are long-ranged and do not decay with inter-qubit distance.

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arXiv2021

Experimental Quantum Learning of a Spectral Decomposition

Currently available quantum hardware allows for small scale implementations of quantum machine learning algorithms. Such experiments aid the search for applications of quantum computers by benchmarking the near-term feasibility of candidate algorithms. Here we demonstrate the quantum learning of a two-qubit unitary by a sequence of three parameterized quantum circuits containing a total of 21 variational parameters. Moreover, we variationally diagonalize the unitary to learn its spectral decomposition, i.e., its eigenvalues and eigenvectors. We illustrate how this can be used as a subroutine to compress the depth of dynamical quantum simulations. One can view our implementation as a demonstration of entanglement-enhanced machine learning, as only a single (entangled) training data pair is required to learn a 4x4 unitary matrix.

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arXiv2020

Efficient characterization of correlated SPAM errors

State preparation and measurement (SPAM) errors limit the performance of many gate-based quantum computing architecures, but are partly correctable after a calibration step that requires, for an exact implementation on a register of $n$ qubits, $2^n$ additional characterization experiments, as well as classical post-processing. Here we introduce an approximate but efficient method for SPAM error characterization requiring the {\it classical} processing of $2^n \! \times 2^n$ real matrices, but only $O(n^2)$ measurements. The technique assumes that multi-qubit measurement errors are dominated by pair correlations, which are estimated with $n(n-1)k/2$ two-qubit experiments, where $k$ is a parameter related to the accuracy. We demonstrate the technique on the IBM and Rigetti online superconducting quantum computers, allowing comparison of their SPAM errors in both magnitude and degree of correlation. We also study the correlations as a function of the register's geometric layout. We find that the pair-correlation model is fairly accurate on linear arrays of superconducting qubits. However qubits arranged in more closely spaced two-dimensional geometries exhibit significant higher-order (such as 3-qubit) SPAM error correlations.

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arXiv2020

Rigorous measurement error correction

We review an experimental technique used to correct state preparation and measurement errors on gate-based quantum computers, and discuss its rigorous justification. Within a specific biased quantum measurement model, we prove that nonideal measurement of an arbitrary $n$-qubit state is equivalent to ideal projective measurement followed by a classical Markov process $Γ$ acting on the output probability distribution. Measurement errors can be removed, with rigorous justification, if $Γ$ can be learned and inverted. We show how to obtain $Γ$ from gate set tomography (R. Blume-Kohout et al., arXiv:1310.4492) and apply the error correction technique to single IBM Q superconducting qubits.

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arXiv2020

Efficient correction of multiqubit measurement errors

State preparation and measurement (SPAM) errors limit the performance of near-term quantum computers and their potential for practical application. SPAM errors are partly correctable after a calibration step that requires, for a complete implementation on a register of $n$ qubits, $2^n$ additional measurements. Here we introduce an approximate but efficient method for multiqubit SPAM error characterization and mitigation requiring the classical processing of $2^n \! \times 2^n$ matrices, but only $O(4^k n^2)$ measurements, where $k=O(1)$ is the number of qubits in a correlation volume. We demonstrate and validate the technique using an IBM Q processor on registers of 4 and 8 superconducting qubits.

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