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An approximate description of quantum states
Marco Paini, Amir Kalev
TL;DR
Reconstructing an N-qubit density operator requires exponentially many numbers, so the paper instead describes states through reusable measurement snapshots. This enables observable expectation-value estimates with error controlled by the observable seminorm and independent of N, and supports a variational optimization method.
Problem
Reconstructing a finite N-qubit density operator requires exponentially many numbers, making density-matrix estimation and storage difficult.
Source: §I · p. 1Method
The paper represents an approximate quantum state by a set of measurement snapshots that can be used to calculate expectation values of observables.
Source: §III · p. 4Results
The approximate description provides an operational procedure whose expectation-value error depends on snapshot count and observable seminorm, independently of N.
Source: B. N qubits · p. 4, §VI · p. 8Takeaways & Limitations
The same measurements can support expectation-value calculations across observable variations in a variational optimization, without a new measurement set for each variation.
Source: §IV · p. 6, §V · p. 7
Abstract
from arXiv · showhide
We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of $N$. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, $N$-independent, seminorm of the observables. Building the approximate description of the $N$-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements and can be considered for implementation on today's Noisy Intermediate-Scale Quantum (NISQ) computers. The access to the expectation values of all observables for a given state leads to an efficient variational method for the determination of the minimum eigenvalue of an observable. The method represents one example of the practical significance of the approximate description of the $N$-qubit state. We conclude by briefly discussing extensions to generative modelling and with fermionic operators.
I. INTRODUCTION
Density-matrix reconstruction scales exponentially with the number of qubits, motivating a description that preserves access to observable expectation values without that representation cost. The paper proposes estimating those values with a copy count independent of N, with error depending on an observable seminorm.
I. INTRODUCTION
- Reconstructing a finite N-body density operator requires exponentially many numbers, making estimation and storage difficult even for systems of a few tens of qubits.Earlier approaches reduced measurement or storage requirements in particular settings, but the dependence on N for generic states remained an open question.Source: §I · p. 1
- Prior work reduced resources for specific tasks, including tomography, estimating selected observables, storing low-bond-dimension states, and approximate descriptions from stabilizer-state inner products.The remaining question was whether dependence on N could be significantly reduced or eliminated for a general protocol.Source: §I · p. 1
- The paper answers affirmatively by showing that any observable’s expectation value can be estimated to arbitrary precision with a number of copies independent of N, but dependent on an observable seminorm.The resulting approximate state description avoids density matrices’ exponential growth in N.Source: §I · pp. 1–2
- The approach reframes state description around expectation values, using group-theoretic tomography to obtain them rather than reconstructing the full density operator.The introduction motivates this choice by noting that expectation values can be estimated from finitely many copies even in some infinite-dimensional systems, with requirements depending on observable properties.Source: §I · pp. 1–2
A. One qubit
The one-qubit construction uses SU(2) tomography to estimate observable expectation values from randomized measurement directions and single-qubit measurements. It builds estimators from identity and Pauli operators and introduces an observable seminorm for characterizing them.
A. One qubit
- SU(2) supplies an irreducible unitary representation for a one-qubit system, providing the group-theoretic basis for the tomography construction.The subsection derives one-qubit estimators and an upper bound for their variance as a foundation for the N-qubit generalization.Source: A. One qubit · p. 2
- Linearity lets the generic observable be expanded in the identity and Pauli operators, whose estimators and variances are evaluated to construct the one-qubit estimator.The variance argument uses that a random variable’s variance is at most its second moment, with the Pauli estimator’s second moment given as 3.Source: A. One qubit · pp. 2–3
- The subsection introduces an observable seminorm to characterize the estimator behavior for a generic observable.The supplied passage states that the seminorm is defined and that its properties are proved in the appendix.Source: A. One qubit · p. 3
- Each sample chooses a direction, rotates the qubit, measures in the computational basis, and uses the result to calculate an estimator for O.Repeating this procedure and averaging the estimators yields an approximation to the expectation value.Source: A. One qubit · p. 3
B. N qubits
The N-qubit method extends the one-qubit tomography to distinguishable qubits with qubit-specific measurement capability, using local rotations and measurements. Its observable-dependent seminorm controls statistical error, while snapshots can be reused to estimate different observables.
B. N qubits
- For distinguishable qubits with qubit-specific measurement capability, the method generalizes the one-qubit tomography to N qubits.It takes the system space as a tensor product and uses SU(2)^N with a tensor-product representation.Source: B. N qubits · p. 3
- Tensor-product observables have estimators formed as products of the corresponding single-qubit estimators, and the general observable is expanded in Pauli strings.The estimator variance has an upper bound, while the seminorm weights Pauli-coefficient pairs according to their overlap structure.Source: B. N qubits · pp. 3–4
- The statistical error is bounded by the observable’s seminorm and the number of preparations, rather than by N alone.An observable’s seminorm is unchanged when the same physical observable is extended by identities, although an N-fold Pauli product can have seminorm 3N/2 as printed in the passage.Source: B. N qubits · p. 4
- The seminorm squared is generally a good approximation of an upper bound on estimator variance, and can be substantially smaller than the seminorm itself.For a chosen statistical error, the required number of identical preparations is directly proportional to estimator variance.Source: B. N qubits · p. 4
- A tomographic sample consists of independently sampled local directions and measurement outcomes, which can be stored as a snapshot and reused to estimate any given observable.The procedure uses single-qubit rotations followed by computational-basis measurements and is described as suitable for NISQ computers.Source: B. N qubits · p. 4
III. THE APPROXIMATE QUANTUM STATE
The paper defines an approximate quantum state as a set of snapshots that can estimate any observable’s expectation value. Its estimation error is independent of N, while snapshot storage grows linearly with N.
III. THE APPROXIMATE QUANTUM STATE
- The approximate state is a set of M independent snapshots that can be stored and reused to estimate any observable’s expectation value.Each snapshot records sampled angles and measurement outcomes from one execution of the tomographic procedure.Source: §III · p. 4
- The estimation error is independent of N and bounded in terms of the observable’s seminorm and the number of snapshots.The passage states a bound proportional to ∥O∥/M for sufficiently large M.Source: §III · p. 4
- Snapshot size grows linearly with N, while the approximation depends only on the observable and the number of snapshots.There are infinitely many approximate states of a given cardinality, but they are equivalent for calculating expectation values under the stated rule.Source: §III · p. 4
- An operator-valued approximation would retain exponentially many matrix elements and can introduce exponential errors when used for explicit trace calculations.The paper also notes that operator distances depend on the chosen distance measure, whereas expectation-value error is physically relevant to the calculations considered.Source: §III · p. 5
IV. IS SU(2) SPECIAL?
SU(2) is a convenient choice, not a necessary one: Pauli-axis measurements can also define a procedure. However, the Pauli approach is less suitable for generic observables and variational reuse.
IV. IS SU(2) SPECIAL?
- SU(2) is not special; Pauli measurements along x, y, and z can also define an equivalent procedure.The Pauli-based procedure uses an estimator and sampling over Pauli strings.Source: §IV · p. 5
- For observables with only polynomially many nonzero Pauli coefficients, the Pauli scheme is usually ineffective because sampled estimators are highly likely to be zero.The sampling domain has exponential cardinality in N.Source: §IV · p. 5
- The SU(2) procedure is practically more convenient because observables of interest are typically expressed as Pauli expansions.The text contrasts these common expansions with linear combinations of the SU(2) operators.Source: §IV · p. 5
- The Pauli scheme can be made to use prior knowledge of which coefficients are nonzero, but it may then require new measurements for every variational change.The nonzero coefficients can change at each variation; SU(2) tomography instead reuses measurements to evaluate variations through the same approximate state.Source: §IV · pp. 5–6
V. THE APPROXIMATE QUANTUM STATE VARIATIONAL OPTIMISER
The optimiser turns minimum-eigenvalue estimation into classical optimization over a parameterized observable evaluated using an approximate quantum state. Its iterative circuit choices can guarantee non-increasing estimates, but parameter-dependent seminorms and powers of 3 constrain the statistical cost.
V. THE APPROXIMATE QUANTUM STATE VARIATIONAL OPTIMISER
- One tomographic data set for an approximate initial state makes the minimum-eigenvalue search a classical optimization over the parameterized observable, when its estimator is explicitly expressible.The estimator uses O(ξ) = U†(ξ)OU, and the paper notes that this dependence can be difficult to express for arbitrary unitaries.Source: §V · p. 6
- Alternating single-qubit rotations with multi-qubit transformations can explore states both within and across entanglement classes.Single-qubit rotations do not change the entanglement class; the multi-qubit transformations are introduced to reach other classes.Source: §V · pp. 6–7
- Choosing parametric multi-qubit transformations that can equal the identity ensures successive optimized estimates are non-increasing.The construction allows unnecessary transformations to leave the variational state unchanged; the paper also assumes an even N for its example pairing, leaving one qubit out if N is odd.Source: §V · pp. 6–7
- A final approximate-state estimate of the prepared state bounds statistical error by ∥O∥ divided by the square root of its cardinality, independent of ξ.The seminorm of the transformed observable can still vary during optimization and could lead to suboptimal parameter choices.Source: §V · p. 7
- The optimization's preparation cost can grow rapidly with N because its error bounds contain powers of 3, although common combinatorial cases with P = Q = 2 have factors below ten.For Bravyi–Kitaev-mapped electronic Hamiltonians, the stated overall statistical-error bound grows at most polynomially in N.Source: §V · p. 7
- The optimizer's functional objective can reduce the total number of iterations relative to variational methods that independently evaluate the objective at each parameter choice.The approach can also optimize parametric objective functions by optimizing their parameters alongside the circuit variables.Source: §V · p. 8
VI. CONCLUSIONS AND OUTLOOK
The paper presents an approximate quantum-state description for estimating observables and uses it to introduce a variational optimization method. It points to generative modelling and fermionic tomography as extensions, while noting that hardware tests and further work on optimizer design remain.
VI. CONCLUSIONS AND OUTLOOK
- The constructive approximate-state procedure estimates generic observables with statistical error set by state cardinality and observable seminorm, independent of N, using single-qubit rotations beyond preparations and measurements.The authors identify implementation on quantum hardware and study of noise effects as immediate next steps.Source: §VI · p. 8
- Further theoretical and numerical work is needed to define the optimizer's unitaries and iteration steps.The authors suggest exploring combinations with methods such as VQE.Source: §VI · p. 8
- For bitstring projectors, the approximate-state estimators have variance bounded by 1 with high probability for all N, supporting a proposed extension to Born Machines.The conclusion describes generative modelling as an interesting application for the approximate quantum-state optimizer.Source: §VI · p. 8
- A Grassmann-number tomography extension can estimate fermionic ladder operators directly, avoiding Pauli mappings and their potentially N-growing monomial powers.The proposed snapshots combine sampled real numbers derived from Grassmann numbers with measured electron presence or absence in orbitals.Source: §VI · p. 8
VIII. APPENDICES
The appendix derives the single-qubit estimator from spin measurements and Pauli-operator rotations, then relates its variance to a seminorm of the observable.
A. One qubit
- Rotating Pauli operators changes their direction, allowing the estimator for a Pauli operator to be expressed using the measurement direction.The appendix treats the identity and Pauli operators separately in deriving estimator properties.Source: A. One qubit · p. 10
- The one-qubit estimator is formed as R1[O](m,n) = Tr[OK1(m,n)] from the measurement outcome and direction.The measured spin outcome is along a unit vector n; the text rewrites the spin measurement using Pauli operators with outcomes m = ±1.Source: A. One qubit · p. 10
- The appendix defines an observable seminorm and uses it to bound estimator variance.The variance calculation decomposes the observable into identity and Pauli components.Source: A. One qubit · pp. 10–11
B. N qubits
For N qubits, the appendix bounds estimator variance using observable seminorms and examines when a simpler seminorm approximates the standard deviation, especially for random states and projectors.
B. N qubits
- For random observables or states, mixed-sign terms and smaller higher-weight Pauli expectations make ∥O∥2 generally a good approximation of the estimator's standard deviation.The appendix motivates examining pure states and computational-basis projectors as a significant special case.Source: B. N qubits · pp. 12–14
- The appendix evaluates mixed-term averages over density operators by averaging unitary transformations and then examines their variance over pure states.It notes that a general upper bound on the variance expression may be weak because it depends strongly on O.Source: B. N qubits · pp. 13–14
- For any P, Equation (65) bounds Var(R[P]) for a random state by ∥P∥2^2 < 1, with probability exponentially close to 1 as a function of N.Source: B. N qubits · p. 14
C. Pauli tomography
Pauli tomography estimates observable expectation values from measurement outcomes, with sampling restricted to relevant coefficients and chosen to control estimator variance.
C. Pauli tomography
- Equation (25) is derived by tracing the density-operator relation after multiplying by the density operator and evaluating in the Pauli-monomial eigenbasis.Source: C. Pauli tomography · p. 14
- Restricting the sum to indices with nonzero observable coefficients avoids projections that would otherwise contribute null terms.The method defines LO as the set of indices with nonzero coefficients and rewrites the estimator over that set.Source: C. Pauli tomography · pp. 14–15
- The estimator weights measurement outcomes by their probabilities and normalizes for uniform sampling over the relevant indices.The text defines p(mi1, . . . , miN) as the outcome probability and 1/|LO| as the normalization under uniform sampling.Source: C. Pauli tomography · p. 15
- Sampling indices from a distribution ˜p(i) changes the estimator to use the corresponding conditional outcome probability and inverse-probability weight.The generalized expression is given for sampling from ˜p(i) over LO.Source: C. Pauli tomography · p. 15
- Choosing the sampling distribution with exponent x = 1 minimizes the stated upper bound on estimator variance.The passage identifies the expectation of the estimator's square as an upper bound for variance.Source: C. Pauli tomography · p. 15
D. Seminorm changes under unitary transformations
The seminorm's behavior under unitary conjugation depends on whether the transformation preserves Pauli degree: product single-qubit unitaries do, while two-qubit gates can mix degree blocks.
D. Seminorm changes under unitary transformations
- The operator space is decomposed into subspaces Bk spanned by Pauli monomials with the same number of nonidentity Pauli factors.The unitary action is defined by conjugation, U(O) = U†OU.Source: D. Seminorm changes under unitary… · p. 15
- Product single-qubit unitaries preserve each Pauli-degree subspace, so the seminorm is unchanged under their conjugation.The identity remains unchanged, and each nonidentity single-qubit Pauli operator maps to a linear combination of Pauli operators.Source: D. Seminorm changes under unitary… · p. 16
- Two-qubit gates can mix B1 and B2, changing the squared seminorm by a factor between 1/3 and 3 for the two-qubit case.For N qubits, a layer of N/2 such gates can increase it by at most 3^(P/2), where P is the smaller of the operator's highest Pauli degree and N/2.Source: D. Seminorm changes under unitary… · p. 16