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description= 30 interview questions for quantum computing roles at IBM, Google, IonQ, and more. Covers fundamentals, algorithms, error correction, and Qiskit coding.;
author= QuantumComputingCourses.com;
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Text of the page (random words):
gorithm why does it threaten rsa rsa security rests on the hardness of factoring large integers given n p q finding p and q from n alone is believed to be classically intractable for large n sub exponential but super polynomial with the best known algorithms shor s algorithm solves factoring in polynomial time on a quantum computer the key insight is that factoring reduces to order finding finding the smallest r such that a r 1 mod n for a random a order finding is in turn solved by quantum phase estimation a quantum register is put in superposition over all powers of a mod n and the quantum fourier transform extracts the period r from the resulting interference pattern with high probability classical post processing then uses r to find p and q via greatest common divisors the quantum speedup comes entirely from the qft s ability to extract the period efficiently classically this requires exponential time a fault tolerant quantum computer running shor s algorithm on a 2048 bit rsa key would need roughly 4000 logical qubits and billions of gate operations current machines are far from this but the threat is serious enough that nist has already standardised post quantum cryptographic replacements q13 what is the quantum fourier transform and where is it used the quantum fourier transform qft is the quantum circuit analogue of the discrete fourier transform dft applied to a quantum state x it produces a superposition where the amplitudes encode the fourier coefficients of the input the classical dft on n 2ⁿ points requires o n log n operations with the fast fourier transform the qft on n qubits requires only o n² quantum gates an exponential reduction in gate count the catch is that you cannot efficiently read out all n fourier coefficients measurement collapses the output to one sample the qft is useful when you need to extract a single property of the fourier transformed distribution such as the dominant frequency or period this makes the qft central to period finding algorithms like shor s where the goal is to find the period of a modular exponential function it is also the subroutine inside quantum phase estimation qpe making it foundational to a wide range of quantum algorithms including vqe eigenvalue estimation quantum chemistry simulations and the hhl algorithm for linear systems understanding the qft is a prerequisite for understanding most advanced quantum algorithms q14 what is quantum phase estimation and why is it important quantum phase estimation qpe is an algorithm that estimates the eigenphase of a unitary operator u given an eigenvector u if u u e 2πiφ u qpe outputs a binary approximation of φ with high probability using o 1 ε ancilla qubits for precision ε the algorithm works by using controlled u operations to write the phase information into a quantum register then applying the inverse qft to extract φ as a readable bitstring the controlled applications of u 2 k for increasing k encode φ into the phases of ancilla qubits in a way that the inverse qft can decode qpe is foundational because eigenvalue problems appear throughout science and engineering the energy of a molecule is an eigenvalue of its hamiltonian vqe approximates ground state energies variationally qpe can compute them exactly with sufficient circuit depth and qubits qpe is also the subroutine inside shor s order finding the hhl linear systems algorithm and quantum simulations of chemistry and condensed matter physics almost every exponential quantum speedup for scientific problems traces back to qpe in some form q15 explain qaoa what type of problems does it target the quantum approximate optimisation algorithm qaoa is a hybrid quantum classical algorithm designed to find good approximate solutions to combinatorial optimisation problems problems where you seek the configuration of binary variables that minimises or maximises some objective function examples include maxcut satisfiability k sat portfolio optimisation and scheduling qaoa encodes the problem as a cost hamiltonian h_c whose ground state corresponds to the optimal solution and a mixer hamiltonian h_b that drives transitions between states it prepares an initial equal superposition and then applies alternating layers of e iγh_c and e iβh_b parameterised by angles γ and β a classical optimiser adjusts these angles to maximise the expected value of h_c measured at the end with p layers qaoa is guaranteed to find the exact optimum as p grows large approaching adiabatic quantum computation but performance at small practical p values is problem dependent and not well understood theoretically current research asks whether qaoa on nisq hardware can outperform classical heuristics like simulated annealing for any real problem the answer is not yet clear q16 what is vqe and why do we use a classical optimiser alongside the quantum circuit the variational quantum eigensolver vqe is a hybrid algorithm for estimating the ground state energy of a quantum system primarily targeting quantum chemistry applications like computing molecular binding energies it is based on the variational principle the expectation value of a hamiltonian h with any trial state ψ is always greater than or equal to the true ground state energy so minimising ψ θ h ψ θ over parameters θ approaches the ground state energy the quantum circuit prepares a parameterised trial state ansatz ψ θ and measures the expectation value of the hamiltonian which is decomposed into a sum of pauli operators that can each be measured directly a classical optimiser gradient descent cobyla spsa then updates the parameters θ to reduce the energy estimate the classical optimiser is necessary because finding the optimal circuit parameters is itself an optimisation problem with a complex landscape quantum hardware cannot intrinsically search this space it can only evaluate a single objective value at one set of parameters per circuit execution the hybrid approach offloads the optimisation loop to classical computers keeping quantum circuits shallow enough to run on nisq hardware q17 what is the difference between bqp and np can quantum computers solve np complete problems bqp bounded error quantum polynomial time is the class of decision problems solvable by a quantum computer in polynomial time with error probability at most 1 3 np is the class of problems whose solutions can be verified in polynomial time classically and np complete problems are the hardest problems in np whether p np is the most famous open question in computer science it is widely believed that bqp does not contain all of np while bqp contains some problems outside p like integer factoring which is in bqp but believed to be outside p np complete problems like 3 sat or the travelling salesman problem are not known to be in bqp grover s algorithm gives a quadratic speedup for brute force search but an np complete problem with n 2ⁿ states would still require o 2 n 2 quantum steps still exponential the relationship is p bqp pspace and np is believed to partially overlap with bqp but not be contained in it quantum computers are likely not a silver bullet for np hardness the complexity theoretic evidence suggests the exponential wall is not entirely removed by quantum computation q18 what is quantum simulation and why is it considered one of the most promising near term applications quantum simulation means using a controllable quantum device to mimic the behaviour of another less controllable quantum system the core insight attributed to feynman is that simulating quantum systems classically is generically exponentially hard the state of n interacting particles requires 2ⁿ complex amplitudes a quantum computer with n qubits can naturally represent and evolve this state applications span quantum chemistry computing molecular ground state energies to predict reaction rates and drug behaviour condensed matter physics understanding high temperature superconductors topological materials and materials science designing new battery cathodes or catalysts quantum simulation is considered near term promising because the required circuits are shallower than for full fault tolerant algorithms like shor s molecules of pharmaceutical relevance involve 50 to 200 electrons a scale where even trotterised hamiltonian simulation may become tractable on early fault tolerant machines both vqe and qpe target this application companies like quantinuum ibm and google already demonstrate quantum chemistry simulations on small molecules and the trajectory suggests this will be one of the first areas where quantum computers deliver genuine value 03 error correction and hardware 7 questions open one to read the full answer q19 what is a logical qubit and how does it differ from a physical qubit a physical qubit is an actual hardware qubit a superconducting transmon a trapped ion a spin in silicon etc subject to real noise decoherence and gate errors a logical qubit is an error protected quantum bit encoded across many physical qubits using a quantum error correcting code the logical qubit behaves as a reliable long lived qubit even as individual physical qubits experience errors the encoding works because quantum error correction can detect and fix errors without measuring and thus collapsing the logical state instead it measures stabilisers multi qubit observables that commute with the logical operators but anticommute with error operators to identify where errors occurred without learning the logical information the overhead is significant current estimates suggest 1000 to 10000 physical qubits per logical qubit for codes like the surface code at practically useful error rates depending on the physical error rate a useful fault tolerant algorithm like shor s algorithm on a 2048 bit key requires roughly 4000 logical qubits implying millions of physical qubits this is why the gap between today s noisy hardware and practically useful quantum computation remains large q20 explain the surface code at a high level the surface code is the leading quantum error correcting code for superconducting and similar qubit architectures physical qubits are arranged on a 2d square lattice with data qubits at the vertices and ancilla measurement qubits between them the code encodes one logical qubit across d x d data qubits where d is the code distance the key feature is locality all stabiliser measurements which detect errors involve only nearest neighbour qubits this is crucial for hardware implementation because long range interactions are hard to engineer reliably the stabilisers are products of pauli x and z operators on 2x2 plaquettes by measuring these regularly any single qubit error manifests as a pair of defects violated stabilisers that can be matched and corrected classically using algorithms like minimum weight perfect matching the surface code s threshold error rate below which increasing d reduces logical error rates is approximately 1 which is achievable with current hardware the tradeoff is high physical qubit overhead a distance d code needs roughly 2d² physical qubits for one logical qubit google s 2024 below threshold experiment on its willow processor published in nature was a landmark demonstration of the surface code working as theory predicts q21 what is the quantum error correction threshold theorem the threshold theorem states that if the physical error rate per gate is below a critical threshold value then arbitrarily long quantum computations can be performed with arbitrarily small logical error rate at the cost of polynomial overhead in physical resources the intuition is that error correcting codes can correct errors faster than errors accumulate provided errors are rare enough below the threshold adding more physical qubits increasing the code distance reduces the logical error rate exponentially above the threshold adding more qubits makes things worse because you introduce more errors than you correct the threshold value depends on the code the error model and the correction strategy for the surface code with depolarising noise the threshold is roughly 0 5 to 1 error per gate different codes have different thresholds concatenated codes often have lower thresholds but are more hardware intensive the theorem is foundational because it proves that scalable quantum computing is physically possible in principle it would be impossible if errors always compounded without correction in practice reaching threshold is the primary target for current quantum hardware efforts q22 what are t1 and t2 times and why do they matter t1 energy relaxation time measures how long it takes a qubit excited to 1 to spontaneously decay back to 0 it is an amplitude damping process driven by energy loss to the environment if t1 is short any qubit will probabilistically flip to 0 during a circuit t2 dephasing time or coherence time measures how long a qubit in a superposition state 0 1 2 maintains its phase before the relative phase between 0 and 1 randomises t2 is always less than or equal to 2 t1 due to the contributions of both energy relaxation and pure dephasing pure dephasing t2 is typically limited by low frequency flux noise or charge noise rather than energy loss these times set hard limits on circuit depth a rough rule of thumb is that you can execute on the order of t2 gate time gates before dephasing causes significant errors for superconducting qubits with t2 200 microseconds and gate times 50 nanoseconds that is about 4000 gate operations per qubit improving t1 and t2 is a central goal of hardware research since longer coherence times translate directly into deeper circuits and more complex algorithms q23 what is gate fidelity and how is it measured gate fidelity measures how closely a physical quantum gate implementation matches the ideal unitary operation a fidelity of 1 0 means the gate is perfect a fidelity of 0 99 means a 1 error rate per gate formally the average gate fidelity is the average overlap between the ideal output and the actual output averaged over a uniform distribution of input states the standard measurement technique is randomised benchmarking rb in rb you apply sequences of random clifford gates of increasing length followed by their inverse which should ideally return the qubit to 0 the probability of returning to 0 decays exponentially with sequence length and the decay rate directly gives the error per clifford gate this technique is robust to state preparation and measurement spam errors which would otherwise contaminate direct fidelity measurements process tomography is a more complete characterisation it reconstructs the full quantum process matrix but requires resources that grow exponentially with the number of qubits and is only practical for one or two qubits interleaved randomised benchmarking extends rb to characterise a specific gate by interleaving it with random cliffords current two qubit gate fidelities are around 99 0 to 99 9 for leading superconducting and...
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