Code Review for Readability and Performance

CereCore provides IT services for hospital systems, including a patient lookup service used by front-desk staff. New engineers on the team say this class takes them far too long to understand.

public class PatientLookupService {
    private List<Patient> a;

    public Patient findByMrn(String x1) {
        for (int i = 0; i < a.size(); i++) {
            Patient tmp = a.get(i);
            for (int j = 0; j < a.size(); j++) {
                if (a.get(j).getMrn().equals(x1) && a.get(j).getStatus().equals("ACTIVE")) {
                    return a.get(j);
                }
            }
        }
        return null;
    }

    public Patient findByName(String name) {
        String query = "SELECT * FROM patients WHERE name = '" + name + "' AND status = 'ACTIVE'";
        return executeQuery(query);
    }
}

Review this snippet and suggest improvements for readability and performance. Name specific problems and explain how you would fix each one.