Light-Based Quantum Computing Hits Fundamental Limits for State Creation

Scientists are continually seeking methods to generate discrete-variable quantum states using linear optical elements and single-photon detection. Deepesh Singh, Ryan J. Marshman, and Luis Villegas-Aguilar, from the Centre for Quantum Computation and Communication Technology at The University of Queensland, alongside Jens Eisert and Nora Tischler, have now developed a rigorous approach to determine the fundamental limits of heralded state generation. Their research addresses a key challenge in photonics, the absence of strong non-linearities, by applying algebraic geometry techniques, specifically the Nullstellensatz Linear Algebra algorithm, to definitively prove when a desired quantum state cannot be created with a given setup. This capability represents a significant advance, allowing researchers to establish lower bounds on the resources required for photonic quantum information processing and validate existing protocols.

Scientists are addressing a major challenge in photonic quantum technologies, which involves developing strategies to prepare suitable discrete-variable quantum states using simple input states, linear optics, and auxiliary photon measurements to identify successful outcomes. Fundamentally, this challenge arises from the lack of strong non-linearities on the single-photon level. Photonic state preparation based on linear optics cannot benefit from the deterministic gate-b.

Applying algebraic geometry to photonic quantum state preparation feasibility

Scientists are increasingly focused on quantum state generation using probabilistic methods. Instead of a deterministic approach available to other physical platforms, quantum states can be probabilistically implemented using single photons, linear-optical networks, and photon detection. They demonstrate this capability to validate and establish lower bounds on the physical resource requirements for several ubiquitous optical states and gates.

Entangled photonic states are crucial resources in quantum computing, sensing, cryptography, and communication. The deterministic generation of entangled states from separable states would require strong non-linearities, but implementing such non-linearities in practical optical platforms remains extremely challenging.

By contrast, linear optics is experimentally accessible, but it permits only a limited set of deterministic transformations in the space of photonic quantum states for any reasonable encoding. Various approaches have been developed to formally characterise the limitations of linear optics for deterministic transformations.

However, arbitrary state transformations can be implemented probabilistically using a combination of single photons, linear optics, and photon detection, provided access to sufficiently many photons. This is particularly relevant in the quest to create linear-optical quantum computers and simulators in a measurement-based fashion.

This work focuses on fundamental limitations, no-go results, associated with probabilistic transformations in linear-optical systems when a limited number of photons are available. The probabilistic nature of photonic state generation necessitates the identification of successful events. There are two general approaches: post-selected schemes, where successful events are sifted from the data set by measuring the photons that make up the target state, and heralded schemes, where only auxiliary photons in ancillary modes are measured to provide an independent signal that the target state has been created.

Unlike post-selection, where the target state is destroyed upon identification, heralding allows the quantum state to be utilised freely, without restrictions on subsequent interference of modes, and allows multiplexing to increase the probability of success. This makes heralded state generation schemes far more useful, and they are the focus of this article.

The feasibility of a state generation task, determining whether an input state can be transformed into a target state using a given measurement pattern, is generally a difficult problem. It can be reformulated as deciding whether a corresponding system of polynomial equations possesses a solution. Previous work has proposed the use of Gröbner basis techniques from algebraic geometry to find solutions to state generation tasks, similar to new approaches using algebraic geometry for quantum error correction.

This approach determines the feasibility of state generation tasks and provides the solutions, i.e., suitable transformations. However, computing a Gröbner basis can be computationally expensive, with worst-case time complexity being doubly exponential. Furthermore, it is progress-free, so the full calculation must be completed to obtain any results.

The Gröbner basis approach may be excessive when only the feasibility of a state generation task needs to be determined, without necessarily finding a specific linear-optical circuit. If a certificate is found, it provides definitive proof that the system of polynomial equations has no solutions, thus attesting to the impossibility of generating the target state using a given input state and measurement scheme.

This rigorous proof contrasts with typical numerical searches, which may suspect but cannot know for sure that no solution exists. The crucial advantage of algebraic geometry approaches is the rigorous nature of the results obtained. Since NulLA focuses solely on deciding the feasibility of the system without finding the solution itself, it is anticipated to be particularly useful in such decision problems.

These problems are directly linked to resource requirements. The approach allows proving that the generation of a given target state is impossible when starting with a certain number of single photons, providing lower bounds on resource requirements and insights into problems without known solutions. Although the focus is on generating photonic quantum states from deterministic single-photon sources, the technique can also be extended to handle probabilistic photon-pair sources, as realised with non-linear processes such as spontaneous parametric down-conversion and four-wave mixing.

Moreover, it can be extended from photonic quantum state generation to photonic quantum gates. The rest of the article is structured as follows: Section II explains the mapping of the heralded state generation task to solving a system of polynomial equations and proving its infeasibility using the NulLA algorithm.

Section III outlines the scaling of the technique and discusses simplifications applicable to the input state and measurement pattern, enabling categorization based solely on the number of available input photons. Section IV highlights example applications of the technique and Section V contains concluding remarks.

In this section, researchers revisit how a heralded photonic quantum state generation task can be cast as a system of polynomial equations, as first described in previous work. They then propose applying the NulLA algorithm for a targeted approach to deciding the feasibility of heralded photonic state preparation tasks.

A. Formulation of a heralded photonic state generation task as a system of polynomial equations. This subsection provides an overview of the steps presented in previous work to determine a linear transformation that probabilistically evolves an input state to a target state conditional on measuring a heralding pattern.

The linear transformation is treated as unknown, while the input state, target state, and heralding pattern are assumed to be given. The linear transformation is not forced to be unitary initially because this choice makes the problem amenable to convenient algebraic geometry tools and unitarity can be recovered later by rescaling and extending its dimensions through the inclusion of ancillary modes.

In the approach taken, the input state vector |ψin⟩ is first transformed into |ψout⟩ via the linear transformation represented by the matrix A. From |ψout⟩, an output state vector |ψpost⟩ is created by conditioning on a particular measurement pattern in the heralding modes. For the photonic quantum state generation task, this output state needs to be equivalent, equal up to a nonzero scaling factor, to the target state vector |ψtar⟩.

All multi-mode states with a fixed photon number can be written as homogeneous polynomials of creation operators, of degree equal to the total number of photons and with the number of variables equal to the number of modes in the corresponding state. The equivalence of any two states can then be reduced to the equivalence of the corresponding two polynomials of creation operators.

Equating the corresponding monomial coefficients results in the ability to solve for the transformation. The input state to the transformation is assumed to be a product state and hence its corresponding polynomial R can be represented by a single monomial |ψin⟩:= R(a†1,in, . , a†N,in)|0⟩ = ∏i=1N 1/√ni.a†ni,in|0⟩.

By rescaling if necessary, one can ensure that a linear transformation A has spectral norm ∥A∥≤3. Then A is unitary if all singular values equal one, and otherwise unitarity can be recovered later by embedding it in a larger transformation. Thus, one can consider A as a general linear transformation without assumptions on its singular values.

The creation operators for the modes 1 ≤i ≤N evolve as a†i,in 7→∑j=1NAi,ja†j, where the subscript “in” indicates input modes, while no subscript indicates output modes. The input state vector evolves as |ψout⟩:= F(a†1, . , a†N)|0⟩ = ∏i=1N 1/√ni. ∑j=1NAi,ja†j ni |0⟩, where F(a†1, . , a†N) is a homogeneous polynomial of degree n in the creation operators a†i for 1 ≤i ≤N.

The coefficients of the monomials in F are themselves homogeneous polynomials of degree n belonging to the polynomial ring C[Ai,j]. The effect of measuring the last M modes of the linear network and heralding on the photon measurement pattern (m1, . , mM) results in the heralded state vector |ψpost⟩:= G(a†1, . , a†N−M)|0⟩ =⟨m1, m2, . , mM|ψout⟩ = 1/∏i=1M(mi. ) ∂F(a†1, . , a†N)/∂a†m1(N−M+1)∂a†m2(N−M+2) . ∂a†mM N a†(N−M+1)=0 . a†(N)=0 |0⟩, where G(a†1, . , a†N−M) is a homogeneous polynomial of degree n−m in the creation operators a†i, here 1 ≤i ≤(N −M).

The coefficients of the monomials in G are again homogeneous polynomials of degree n belonging to the polynomial ring C[Ai,j]. The target state vector can also be represented as a homogeneous polynomial |ψtar⟩:= Q(a†1, . , a†N−M)|0⟩. The equivalence of the heralded state from the circuit and the target state can hence be reduced to the equivalence of the corresponding polynomials. This work focuses on proving the infeasibility of generating target states from given input states and measurement schemes, offering a rigorous alternative to numerical searches.

The NulLA algorithm serves as a progressive method for identifying infeasibility certificates, with increasing computational complexity corresponding to higher degrees of certainty. Specifically, the study demonstrates the capability to validate and establish lower bounds on physical resources needed for ubiquitous states and gates in quantum technologies.

The technique maps the heralded state generation task to solving systems of polynomial equations, allowing for definitive proof of impossibility when no solutions exist. Unlike traditional approaches, this method provides rigorous results, confirming the absence of solutions rather than merely failing to find them through numerical searches.

The research extends beyond single-photon sources to encompass probabilistic photon-pair sources, such as those created via spontaneous parametric down-conversion and four-wave mixing. Furthermore, the methodology is applicable not only to photonic quantum state generation but also to photonic quantum gates, broadening its potential impact.

By focusing on feasibility decisions, NulLA anticipates improved efficiency in determining resource requirements for state generation tasks. This approach allows researchers to prove that generating a specific target state is impossible with a limited number of photons, establishing lower bounds on necessary resources and offering insights into unsolved problems.

Algebraic geometry confirms limits to photonic quantum state generation

Scientists have developed a rigorous method for determining the feasibility of creating specific quantum states using linear optics and photon detection. The research demonstrates the application of this algorithm to several common quantum states and gates, validating known limitations and establishing lower bounds on the resources required for their creation.

The method accounts for the probabilistic nature of photonic sources, which often emit unwanted photons that reduce the fidelity of the target state, and extends beyond state generation to the design of heralded gates. While determining feasibility can be computationally demanding, the study reveals that proving infeasibility often requires testing only low-degree certificates, offering a practical advantage.

A key finding is the ability to identify fundamental limitations in optical setups and measurement schemes, providing valuable insights into the resource requirements for quantum optical state generation and manipulation. The authors acknowledge that the computational complexity of proving feasibility remains a challenge.

Future work will likely focus on refining the algorithm and applying it to more complex quantum systems, contributing to the advancement of linear-optical quantum computing by clarifying the boundaries of what is achievable with current technologies. A MATLAB code implementing the algorithm is publicly available to facilitate further research in this area.