Introduction

While near-term quantum devices are approaching the limits of classical tractability1,2, they are limited in the number of physical qubits, and can only execute finite-depth circuits with sufficient fidelity to be useful in applications3. Hybrid quantum-classical algorithms hold great promise for achieving a meaningful quantum advantage in the near-term4,5,6 by combining the quantum resources offered by near-term devices with the computational power of classical processors. In a hybrid algorithm, a classical computer preprocesses a problem instance to expose the core components which capture its essential computational hardness, and is also in a form compatible with a quantum algorithm. The classical computer then utilizes a near-term quantum computer to finish solving the problem. The output of the quantum computer is a set of classical bitstrings which are post-processed classically. This interaction may be iterative: the classical computer sends communication and control signals to and from the quantum device, for example, by proposing new parameter values to be used in a parameterized quantum circuit.

Quantum algorithms developed for near-term quantum hardware will have to contend with several hardware restrictions, such as limited number of qubits, coupling connectivity, gate fidelities, and various sources of noise, which are all highly dependent on the particular quantum processor that the algorithm is being run on. Trade offs between different resources have recently been demonstrated in the context of running quantum circuits7. However, even as fault-tolerant quantum processors are built, classical computing will still be required to perform preprocessing and post-processing, as well as for quantum error-correction8. Therefore, understanding how quantum and classical computing resources can be leveraged together is imperative for extracting maximal utility from quantum computers.

An illustration of this hybrid scheme, in the context of this work, is shown in Fig. 1. In this work, we implement the variational quantum factoring (VQF) algorithm9 on a superconducting quantum processor10. VQF is an algorithm for tackling an NP problem with a tunable trade off between classical and quantum computing resources, and is feasible for near-term devices. VQF uses classical preprocessing to reduce integer factoring to a combinatorial optimization problem which can be solved using the quantum approximate optimization algorithm (QAOA)11.

VQF is a useful algorithm to study for several reasons. First, an inverse relationship exists between the amount of classical preprocessing and the number of required qubits needed for solving the problem, resulting in a tunable trade off between the amount of classical computation and qubits used. Second, the complexity of the quantum circuit acting on those qubits is tunable by adjusting the number of layers used in the QAOA ansatz. Finally, VQF has the advantage that its success is quickly verifiable on a classical computer by checking if the proposed factors multiply to the input biprime.

Results

Variational quantum factoring

The VQF algorithm maps the problem of factoring into an optimization problem12. Given an n-bit biprime number

$$N=\mathop{\sum }\limits_{k=0}^{n-1}{2}^{k}{N}_{k},$$
(1)

factoring involves finding the two prime factors p and q satisfying N = pq, where

$$p=\mathop{\sum }\limits_{k=0}^{{n}_{p}-1}{2}^{k}{p}_{k},$$
(2)
$$q=\mathop{\sum }\limits_{k=0}^{{n}_{q}-1}{2}^{k}{q}_{k}.$$
(3)

That is, p and q can be represented with nq and np bits, respectively.

Factoring in this way can be thought of as the inverse problem to the longhand binary multiplication: the value of the ith bit of the result, Ni, is known and the task is to solve for the bits of the prime factors {pi} and {qi}. An explicit binary multiplication of p and q yields a series of equations that have to be satisfied by {pi} and {qi}, along with carry bits {zi,j} which denote a bit carry from the ith to the jth position. Re-writing each equation in the series allows it to be associated to a particular clause in an optimization problem relating to the bit Ni:

$${C}_{i}={N}_{i}-\mathop{\sum }\limits_{j=0}^{i}{q}_{j}{p}_{i-j}-\mathop{\sum }\limits_{j=0}^{i}{z}_{i,j}+\mathop{\sum }\limits_{j=1}^{{n}_{p}+{n}_{q}-1}{2}^{j}{z}_{i,i+j}.$$
(4)

In order for each clause, Ci, to be 0, the values of {pi}, {qi} and the carry bits {zi,j} in the clause must be correct. By satisfying the constraint for all clauses, the factors p and q can be retrieved.

Thus, the problem of factoring is reduced to a combinatorial optimization problem. Such problems can be solved using quantum computers by associating a qubit with each bit in the cost function. The number of qubits needed depends on the number of bits in the clauses. As discussed in9, some number of classical preprocessing heuristics can be used to simplify the clauses (that is, assign values to some of the bits {pi} and {qi}). As classical preprocessing removes variables from the optimization problem (by explicitly assigning bit values), the number of qubits needed to complete the solution to the problem is reduced. The new set of clauses will be denoted as $$\{{C}_{i}^{\prime}\}$$ (see Supplementary Material for details of the classical preprocessing).

Each clause $${C}_{i}^{\prime}$$ can be mapped to a term in an Ising Hamiltonian $${\hat{C}}_{i}$$ by associating each bit value (bk {pi, qi, zi,j}) to a corresponding qubit operator:

$${b}_{k}\mapsto \frac{1}{2}(1-{Z}_{k}).$$
(5)

The solution to the factoring problem corresponds to finding the ground state of the Hamiltonian

$${\hat{H}}_{{C}^{\prime}}=\mathop{\sum }\limits_{i=0}^{n}{\hat{C}}_{i}^{2},$$
(6)

with a well defined ground state energy E0 = 0. Because each clause Ci in equation (4) contains quadratic terms in the bits and the Hamiltonian $${\hat{H}}_{{C}^{\prime}}$$ is a sum of squares of $${\hat{C}}_{i}$$, $${\hat{H}}_{{C}^{\prime}}$$ includes up to 4-local terms of the form ZiZjZkZl. (An operator is k-local if it acts non-trivially on at most k qubits.) Much of the literature on quantum optimization has considered solving MAXCUT problems on d-regular graphs11,13, which can be mapped to problems of finding the ground states of 2-local Hamiltonians. Other problems, such as MAX-3-LIN-214,15, can be directly mapped to ground state problems of 3-local Hamiltonians. The 4-local Ising Hamiltonian problems produced for VQF motivates a new entry to the classes of Ising Hamiltonian problems currently studied in the context of QAOA. While any k-local Hamiltonian can be reduced to a 2-local Hamiltonian by well-known techniques16,17, the interactions between the qubits in the resulting 2-local Hamiltonian is not guaranteed to correspond to a d-regular graph, which again falls outside the scope of existing considerations in the literature.

The ground state of the Hamiltonian in equation (6) can be approximated on near-term, digital quantum computers using QAOA11. QAOA is a promising candidate for demonstrating quantum advantage through combinatorial optimization18,19,20. Each layer of a QAOA ansatz consists of two unitary operators, each with a tunable parameter. The first unitary operator is

$${U}_{H}(\gamma )={e}^{-i\gamma {\hat{H}}_{{C}^{\prime}}}\,\,{{\mbox{where}}}\,\,\gamma \in [0,2\pi ),$$
(7)

which applies an entangling phase according to the cost Hamiltonian $${\hat{H}}_{{C}^{\prime}}$$. The second unitary is the admixing operation

$${U}_{a}(\beta )=\mathop{\prod }\limits_{i}^{n}{e}^{-i\beta {X}_{i}}\,\,{{\mbox{where}}}\,\,\beta \in [0,\pi ),$$
(8)

which applies a single-qubit rotation around the X-axis with angle 2β to each qubit.

For a given number of layers p, the combination of UH(γ) and Ua(β) are repeated sequentially with different parameters, generating the ansatz state

$$\left|{{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}}\right\rangle ={U}_{a}({\beta }_{p-1}){U}_{H}({\gamma }_{p-1})\cdots {U}_{a}({\beta }_{0}){U}_{H}({\gamma }_{0}){\left|+\right\rangle }^{\otimes n},$$
(9)

parameterized by γ = (γ0,  , γp−1) and β = (β0, , βp−1). The approximate ground state for the cost Hamiltonian can reached by tuning these 2p parameters. Given this ansatz state, a classical optimizer is used to find the optimal parameters γopt and βopt that minimize the expected value of the cost Hamiltonian

$$E({{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}})=\left\langle {{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}}\right|{\hat{H}}_{{C}^{\prime}}\left|{{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}}\right\rangle ,$$
(10)

where for different parameters γ and β the value E(γ, β) is estimated on a quantum computer.

The circuit generating the state $$\left|{{{{\boldsymbol{\gamma }}}}}_{{{{\rm{opt}}}}},{{{{\boldsymbol{\beta }}}}}_{{{{\rm{opt}}}}}\right\rangle$$ is then prepared on the quantum computer and measured. If the outcome of the measurement satisfies all the clauses, it can be mapped to the remaining unsolved binary variables in {pi}, {qi}, and {zi,j}, resulting in the prime factors p and q.

The success of VQF is measured by the probability that a measured bitstring encodes the correct factors. We define the set Ms = {mj} consisting of all bitstrings sampled from the quantum computer that satisfy all the clauses in the Hamiltonian, $${\hat{H}}_{{C}^{\prime}}$$. We can therefore define the success rate, s(γ, β), as the proportion of the bitstrings sampled from $$\left|{{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}}\right\rangle$$ that satisfy all clauses in $$\{{C}_{i}^{\prime}\}$$:

$$s({{{\boldsymbol{\gamma }}}},{{{\boldsymbol{\beta }}}})=\frac{\left|{M}_{s}\right|}{\left|M\right|},\ {M}_{s}=\{{m}_{j}\in M| \sum {C}_{i}=0\}$$
(11)

where $$\left|{M}_{s}\right|$$ is the number of samples satisfying all the clauses and $$\left|M\right|$$ is the total number of measurements sampled.

In order to successfully factor the input biprime, only one such satisfactory bitstring needs to be observed. However, if that bitstring occurs very rarely, then many repeated preparations and measurements of the trial state are necessary. Therefore, a higher value of s(γ, β) is generally preferred. E(γ, β) ≥ 0, with the equality condition satisfied if s(γ, β) = 1.

Experimental analysis

In this work, we implement the VQF algorithm using the QAOA ansatz on the ibmq_boeblingen superconducting quantum processor (see Supplementary Material) to factor three biprime integers: 1099551473989, 3127, and 6557 which are classically preprocessed to instances with 3, 4, and 5 qubits respectively. The number of qubits required for each instance varies based the amount of classical preprocessing performed; 24 iterations of classical preprocessing were used for 1099551473989, 8 iterations for 3127, and 9 iterations for 6557.

In order to optimize the QAOA circuit and find γopt and βopt we employ a layer-by-layer approach21. This approach has two phases: In the first phase, for each layer k, we evaluate a two-dimensional energy landscape by sweeping through different γk and βk values while fixing the parameters for the previous layers (1 through k − 1) to the optimal values found in previous iterations. In our experiments, for 1099551473989 we evaluate the energy for discrete values of γk and βk ranging from 0 to 2π with a resolution of π/6, which yields 144 circuit evaluations for each layer (Fig. 2). For 3127 and 6557, we use a resolution of 2π/23, which yields 529 circuit evaluations for each layer (see Supplementary Material). We use the degeneracy in the energy landscape as a result of evaluating β up to 2π instead of π as a feature to better understand the pattern of the energy landscape. We then select the optimal values for γk and βk from this grid, and combine them with the optimal values from the previous layer. We increment k until it reaches the last layer, in which case the completion of the final round of the above steps marks the end of the first phase of the algorithm. In the second phase, we use the values {(γk, βk)} obtained from the first phase as the initial point for a gradient-based optimization using the L-BFGS-B method over all 2k parameters. Although there are alternative strategies for training QAOA circuits21,22, this approach has been shown to require a grid resolution that scales polynomially with respect to the number of qubits required9.

In order to measure the gradient of E(γ, β) with respect to the individual variational parameters, we use analytical circuit gradients using the parameter-shift rule23. We use the optimal parameters found for a circuit with p layers using this method to prepare and measure $$\left|{{{{\boldsymbol{\gamma }}}}}_{{{{\rm{opt}}}}},{{{{\boldsymbol{\beta }}}}}_{{{{\rm{opt}}}}}\right\rangle$$ in order to calculate the algorithm success rate (Fig. 3).

The performance of our algorithm relies on the optimization over an energy surface spanned by the variational parameters γ and β for each layer. Consequently, by studying this energy surface, we can understand the performance of the optimizer in tuning the parameters of the ansatz state, which in turn impacts the success rate. We do so by visualizing the energy landscape for the kth layer of the ansatz as a function of the two variational parameters, βk and γk, fixing (γ0, .., γk−1) and (β0, .., βk−1) to the optimal values obtained through ideal simulation. Comparing these landscapes between experimental results, ideal simulation, and noisy simulation provides us with a valuable insight into the performance of the algorithm on quantum hardware.

The main sources of incoherent noise present in the quantum processor are qubit relaxation and decoherence. In order to capture the effects of these sources of noise on our quantum circuit in simulation, we apply a relaxation channel with relaxation parameter ϵr and a dephasing channel with dephasing parameter ϵd after each gate, described by:

$${{{\mathcal{E}}}}(\rho )=\mathop{\sum }\limits_{m=1}^{3}{E}_{m}\rho {E}_{m}^{{\dagger} },$$
(12)

with Kraus operators

$$\begin{array}{lll}{E}_{1}\,=\,\left(\begin{array}{ll}1&0\\ 0&\sqrt{1-{\epsilon }_{r}-{\epsilon }_{d}}\end{array}\right),\\ {E}_{2}\,=\,\left(\begin{array}{ll}1&\sqrt{{\epsilon }_{d}}\\ 0&0\end{array}\right),\\ {E}_{3}\,=\,\left(\begin{array}{ll}1&0\\ 0&\sqrt{{\epsilon }_{r}}\end{array}\right).\end{array}$$
(13)

Furthermore, a dominant source of coherent error is a residual ZZ-coupling between the transmon qubits24. This interaction is caused by the coupling between the higher energy levels of the qubits, and is especially pronounced in transmons due to their weak anharmonicity. In our experiments, we measure the average residual ZZ-coupling strength to be ξ/2π = 102 ± 124kHz (see Supplementary Material). While the additional ZZ rotation caused by this interaction when the qubits are idle between gates can be compensated by the tunable parameters γ, this interaction has a severe impact on the two-qubit gates. In our experiments the Ising terms eiγZZ = CNOTIZ(γ)CNOT are realized using a single-qubit Z rotation conjugated by CNOT gates. Each CNOT is comprised of a ZX term generated by a two-pulse echo cross-resonance gate25 and single-qubit rotations:

$${U}_{{{{\rm{CNOT}}}}}={e}^{-i\frac{\pi }{4}}{[ZI]}^{-1/2}{[ZX]}^{1/2}{[IX]}^{-1/2}$$
(14)

However, in the presence of residual ZZ-coupling, the axis of rotation of the cross-resonance (CR) gate is altered:

$${{{\rm{CR}}}}(t)={e}^{i{{\Gamma }}ZXt}\mapsto {e}^{i({{\Gamma }}ZX+{{\Xi }})t}$$
(15)

where Γ is the ZX coupling strength between the qubits as the result of the cross-resonance drive and Ξ is the overall effect of the residual ZZ-coupling between the qubits involved in the drive and their neighbors (see Supplementary Material). As a result, the two-qubit interactions are fundamentally altered, which leads to a source of error that cannot be corrected for through the variational parameters.

Figure 2 shows the energy landscapes from factoring 1099551473989 across 3 layers. Comparing the energy landscape produced in experiment with the ideal simulation and noisy simulations indicates that the energy landscapes produced in experiment is in close agreement with those produced by the ideal simulation and the simulation with incoherent noise for p = 1 and p = 2. However, for p = 3 the energy landscapes produced in experiment have considerable deviations from the ideal simulations due to coherent errors. The presence of incoherent sources of noise do not change the energy landscape pattern, and only reduce the contrast at every layer. On the other hand, the residual ZZ-coupling between qubits alters the pattern of the energy landscape. At lower depths, the impact of this noise source is minor, yet as we add more layers of the ansatz the effects accumulate, constructively interfere, and amplify due to the coherent nature of the error. To a first-order approximation, the coherent error contribution leads to changes in the weights of various frequency components of the optimization landscape (see Supplementary Material). In our energy landscapes these errors amplify the weights of high frequency components of the optimization landscape, leading to additional fluctuations compared to the ideal simulation.

We report the average success rate of VQF as a function of the number of layers in the ansatz for experiment and simulation (Fig. 3). As would be expected, in the ideal case the average success rate approaches the ideal value of 1 with increasing layers of the ansatz for each of the instances studied. Comparing experimental results to those obtained from ideal simulation shows a large discrepancy in the success rate. This discrepancy is not explained by the incoherent noise caused by qubit relaxation and dephasing; the coherent error caused by the residual ZZ-coupling is the dominant effect negatively impacting the success rate. Therefore, the performance of VQF, and other QAOA-based hybrid quantum-classical algorithms by extension, is limited by the residual ZZ-coupling between the qubits on transmon-based superconducting quantum processors.

We investigate the impact of different noise channels on the success rate scaling of VQF (Fig. 4). In the ideal scenario, with every added layer of QAOA, which increases the ansatz expressibility26, the algorithm success rate increases. However, in the presence of phase damping, after an initial increase in the success rate we observe a plateau after which the success rate neither increases nor decreases as more layers of the ansatz circuit are added. We expect that this behavior is caused by a loss of quantum coherence at higher layers of circuit, which hinders the quantum interference required for the algorithm. In contrast, while undergoing amplitude damping, after an initial increase in the success rate, due to increased ansatz expressibility, we observe a steady decay in the algorithm’s performance. This decay is a result of qubit relaxation back to the $$\left|0\right\rangle$$ state. When examining the effect of the coherent error induced by the residual ZZ-coupling, we see a more dramatic effect than that induced by either phase damping or amplitude damping. In the presence of the coherent error induced by the residual ZZ-coupling, we observe a plateau in the success rate for lower frequencies (10 and 25 kHz) and a decline in success rate for higher frequencies (50 and 100 kHz) within the first twenty layers of the ansatz.

While each of the sources of noise impact the algorithm’s performance, the residual ZZ-coupling between the qubits in the quantum processor has the most dominant impact on the success rate; Our analysis indicates that this coherent error is the main source of error impeding the effectiveness of the algorithm. The accumulation of coherent noise from this source significantly alters the ansatz at higher layers, and cannot be corrected using the variational parameters. The results shown in Fig. 4 indicate that the performance of VQF can be significantly improved by engineering ZZ suppression27,28 and circuit compilation methods designed for mitigating coherent errors such as randomized compiling29. Since VQF is a non-trivial instance of QAOA we expect this performance analysis to also hold for other quantum algorithms based upon QAOA.

Discussion

In this work, we analyzed the performance of VQF on a fixed-frequency transmon superconducting quantum processor and investigated several kinds of classical and quantum resource tradeoffs. We map the problem of factoring to an optimization problem and use variable amounts of classical preprocessing to adjust the number of qubits required. We then use the QAOA ansatz with a variable number of layers to find the solution to the optimization problem. We find that the success rate of the algorithm saturates as p increases, instead of decreasing to the accuracy of random guessing. While more layers increases the expressibility of the ansatz, at higher depths the circuit suffer from the impacts of noise.

Our analysis indicate that the residual ZZ-coupling between the transmon qubits significantly impacts performance. While relaxation and decoherence have an impact on the quantum coherence, especially at the deeper layers of QAOA, the effect of coherent noise can quickly accumulate and limit performance. By developing a noise model incorporating the coherent sources of noise, we are able to much more accurately predict our experimental results.

There have been various techniques7,30 proposed for benchmarking the capability of a quantum device for performing arbitrary unitary operations. However, recent findings on experimental systems3,31 suggest that existing benchmarking methods may not be effective for predicting the capability of a quantum device for specific applications. This has motivated recent works that focus on benchmarking the performance of a quantum device for applications such as generative modeling32 and fermionic simulation33. Our study of VQF on a superconducting quantum processor has the potential to become another entry in the collection of such application-based benchmarks for quantum computers.